Daily Papers

In the 1990s, the constant error carousel and gating were introduced as the central ideas of the Long Short-Term Memory (LSTM). Since then, LSTMs have stood the test of time and contributed to numerous deep learning success stories, in particular they constituted the first Large Language Models (LLMs). However, the advent of the Transformer technology with parallelizable self-attention at its core marked the dawn of a new era, outpacing LSTMs at scale. We now raise a simple question: How far do we get in language modeling when scaling LSTMs to billions of parameters, leveraging the latest techniques from modern LLMs, but mitigating known limitations of LSTMs? Firstly, we introduce exponential gating with appropriate normalization and stabilization techniques. Secondly, we modify the LSTM memory structure, obtaining: (i) sLSTM with a scalar memory, a scalar update, and new memory mixing, (ii) mLSTM that is fully parallelizable with a matrix memory and a covariance update rule. Integrating these LSTM extensions into residual block backbones yields xLSTM blocks that are then residually stacked into xLSTM architectures. Exponential gating and modified memory structures boost xLSTM capabilities to perform favorably when compared to state-of-the-art Transformers and State Space Models, both in performance and scaling.

In recent years, transformer-based models have gained prominence in multivariate long-term time series forecasting (LTSF), demonstrating significant advancements despite facing challenges such as high computational demands, difficulty in capturing temporal dynamics, and managing long-term dependencies. The emergence of LTSF-Linear, with its straightforward linear architecture, has notably outperformed transformer-based counterparts, prompting a reevaluation of the transformer's utility in time series forecasting. In response, this paper presents an adaptation of a recent architecture termed extended LSTM (xLSTM) for LTSF. xLSTM incorporates exponential gating and a revised memory structure with higher capacity that has good potential for LTSF. Our adopted architecture for LTSF termed as xLSTMTime surpasses current approaches. We compare xLSTMTime's performance against various state-of-the-art models across multiple real-world da-tasets, demonstrating superior forecasting capabilities. Our findings suggest that refined recurrent architectures can offer competitive alternatives to transformer-based models in LTSF tasks, po-tentially redefining the landscape of time series forecasting.

While attention-based architectures, such as Conformers, excel in speech enhancement, they face challenges such as scalability with respect to input sequence length. In contrast, the recently proposed Extended Long Short-Term Memory (xLSTM) architecture offers linear scalability. However, xLSTM-based models remain unexplored for speech enhancement. This paper introduces xLSTM-SENet, the first xLSTM-based single-channel speech enhancement system. A comparative analysis reveals that xLSTM-and notably, even LSTM-can match or outperform state-of-the-art Mamba- and Conformer-based systems across various model sizes in speech enhancement on the VoiceBank+Demand dataset. Through ablation studies, we identify key architectural design choices such as exponential gating and bidirectionality contributing to its effectiveness. Our best xLSTM-based model, xLSTM-SENet2, outperforms state-of-the-art Mamba- and Conformer-based systems of similar complexity on the Voicebank+DEMAND dataset.

Transformers are widely used as generic backbones in computer vision, despite initially introduced for natural language processing. Recently, the Long Short-Term Memory (LSTM) has been extended to a scalable and performant architecture - the xLSTM - which overcomes long-standing LSTM limitations via exponential gating and parallelizable matrix memory structure. In this report, we introduce Vision-LSTM (ViL), an adaption of the xLSTM building blocks to computer vision. ViL comprises a stack of xLSTM blocks where odd blocks process the sequence of patch tokens from top to bottom while even blocks go from bottom to top. Experiments show that ViL holds promise to be further deployed as new generic backbone for computer vision architectures.

Vision Graph Neural Networks (ViGs) offer a new direction for advancements in vision architectures. While powerful, ViGs often face substantial computational challenges stemming from their graph construction phase, which can hinder their efficiency. To address this issue we propose AdaptViG, an efficient and powerful hybrid Vision GNN that introduces a novel graph construction mechanism called Adaptive Graph Convolution. This mechanism builds upon a highly efficient static axial scaffold and a dynamic, content-aware gating strategy called Exponential Decay Gating. This gating mechanism selectively weighs long-range connections based on feature similarity. Furthermore, AdaptViG employs a hybrid strategy, utilizing our efficient gating mechanism in the early stages and a full Global Attention block in the final stage for maximum feature aggregation. Our method achieves a new state-of-the-art trade-off between accuracy and efficiency among Vision GNNs. For instance, our AdaptViG-M achieves 82.6% top-1 accuracy, outperforming ViG-B by 0.3% while using 80% fewer parameters and 84% fewer GMACs. On downstream tasks, AdaptViG-M obtains 45.8 mIoU, 44.8 APbox, and 41.1 APmask, surpassing the much larger EfficientFormer-L7 by 0.7 mIoU, 2.2 APbox, and 2.1 APmask, respectively, with 78% fewer parameters.

We introduce Gated-SwinRMT, a family of hybrid vision transformers that combine the shifted-window attention of the Swin Transformer with the Manhattan-distance spatial decay of Retentive Networks (RMT), augmented by input-dependent gating. Self-attention is decomposed into consecutive width-wise and height-wise retention passes within each shifted window, where per-head exponential decay masks provide a two-dimensional locality prior without learned positional biases. Two variants are proposed. Gated-SwinRMT-SWAT substitutes softmax with sigmoid activation, implements balanced ALiBi slopes with multiplicative post-activation spatial decay, and gates the value projection via SwiGLU; the Normalized output implicitly suppresses uninformative attention scores. Gated-SwinRMT-Retention retains softmax-normalized retention with an additive log-space decay bias and incorporates an explicit G1 sigmoid gate -- projected from the block input and applied after local context enhancement (LCE) but prior to the output projection~W_O -- to alleviate the low-rank W_V !cdot! W_O bottleneck and enable input-dependent suppression of attended outputs. We assess both variants on Mini-ImageNet (224{times}224, 100 classes) and CIFAR-10 (32{times}32, 10 classes) under identical training protocols, utilizing a single GPU due to resource limitations. At {approx}77--79\,M parameters, Gated-SwinRMT-SWAT achieves 80.22% and Gated-SwinRMT-Retention 78.20% top-1 test accuracy on Mini-ImageNet, compared with 73.74% for the RMT baseline. On CIFAR-10 -- where small feature maps cause the adaptive windowing mechanism to collapse attention to global scope -- the accuracy advantage compresses from +6.48\,pp to +0.56\,pp.

Gating mechanisms have been widely utilized, from early models like LSTMs and Highway Networks to recent state space models, linear attention, and also softmax attention. Yet, existing literature rarely examines the specific effects of gating. In this work, we conduct comprehensive experiments to systematically investigate gating-augmented softmax attention variants. Specifically, we perform a comprehensive comparison over 30 variants of 15B Mixture-of-Experts (MoE) models and 1.7B dense models trained on a 3.5 trillion token dataset. Our central finding is that a simple modification-applying a head-specific sigmoid gate after the Scaled Dot-Product Attention (SDPA)-consistently improves performance. This modification also enhances training stability, tolerates larger learning rates, and improves scaling properties. By comparing various gating positions and computational variants, we attribute this effectiveness to two key factors: (1) introducing non-linearity upon the low-rank mapping in the softmax attention, and (2) applying query-dependent sparse gating scores to modulate the SDPA output. Notably, we find this sparse gating mechanism mitigates 'attention sink' and enhances long-context extrapolation performance, and we also release related https://github.com/qiuzh20/gated_attention{codes} and https://huggingface.co/QwQZh/gated_attention{models} to facilitate future research.

Large pretrained models, coupled with fine-tuning, are slowly becoming established as the dominant architecture in machine learning. Even though these models offer impressive performance, their practical application is often limited by the prohibitive amount of resources required for every inference. Early-exiting dynamic neural networks (EDNN) circumvent this issue by allowing a model to make some of its predictions from intermediate layers (i.e., early-exit). Training an EDNN architecture is challenging as it consists of two intertwined components: the gating mechanism (GM) that controls early-exiting decisions and the intermediate inference modules (IMs) that perform inference from intermediate representations. As a result, most existing approaches rely on thresholding confidence metrics for the gating mechanism and strive to improve the underlying backbone network and the inference modules. Although successful, this approach has two fundamental shortcomings: 1) the GMs and the IMs are decoupled during training, leading to a train-test mismatch; and 2) the thresholding gating mechanism introduces a positive bias into the predictive probabilities, making it difficult to readily extract uncertainty information. We propose a novel architecture that connects these two modules. This leads to significant performance improvements on classification datasets and enables better uncertainty characterization capabilities.

Quantum annealers are commercial devices aiming to solve very hard computational problems named spin glasses. Just like in metallurgic annealing one slowly cools a ferrous metal, quantum annealers seek good solutions by slowly removing the transverse magnetic field at the lowest possible temperature. The field removal diminishes quantum fluctuations but forces the system to traverse the critical point that separates the disordered phase (at large fields) from the spin-glass phase (at small fields). A full understanding of this phase transition is still missing. A debated, crucial question regards the closing of the energy gap separating the ground state from the first excited state. All hopes of achieving an exponential speed-up, as compared to classical computers, rest on the assumption that the gap will close algebraically with the number of qspins, but renormalization group calculations predict that the closing will be instead exponential. Here we solve this debate through extreme-scale numerical simulations, finding that both parties grasped parts of the truth. While the closing of the gap at the critical point is indeed super-algebraic, it remains algebraic if one restricts the symmetry of possible excitations. Since this symmetry restriction is experimentally achievable (at least nominally), there is still hope for the Quantum Annealing paradigm.

Retrieval-augmented generation stores all content indiscriminately, degrading accuracy as noise accumulates. Parametric approaches compress knowledge into weights, precluding selective updates. Neither mirrors biological memory, which gates encoding based on salience and archives rather than deletes superseded information. We introduce write-time gating that filters incoming knowledge objects using composite salience scores (source reputation, novelty, reliability) while maintaining version chains that preserve prior states. Using real LLM evaluation without oracle access to quality labels, write gating achieves 100 percent accuracy versus 13 percent for ungated stores. The critical finding emerges under distractor scaling: at 8:1 distractor ratios, read-time filtering (Self-RAG) collapses to 0 percent while write gating maintains 100 percent, revealing a structural advantage of write-time over read-time curation. Validation on Wikipedia (20 entities), procedurally generated pharmacology data, and 2026 arXiv papers confirms these findings. The gating advantage scales inversely with parametric memory support: +25pp for Wikipedia, +48pp for post-cutoff arXiv, +65pp for procedural data with zero training knowledge. Signal ablation confirms the method does not depend on oracle-correlated metadata. Write gating matches Self-RAG accuracy at one-ninth the query-time cost.

Serial verification gates are a core reliability primitive in LLM harnesses: a candidate answer is returned only if k verifier calls all accept it. Under conditionally independent gates, the recent Odds Law (arXiv:2606.15712) shows that posterior log-odds grow linearly in k, so failure decays exponentially, and states that "a tight theory of partially correlated verifier cascades remains open." This note gives a minimal such theory. Modeling the per-instance false-accept rate on the generator's own errors as a latent variable αsim G (de Finetti), the exact cascade posterior is ell_k = ell_0 - ln m_k, with m_k the k-th moment of G. Then: (i) ell_k is concave in k for every non-degenerate G -- the Odds Law is its tangent at the first gate and an upper bound; (ii) for Beta(a,b) latents, failure decays polynomially, 1-r_k asymp k^{-b}, with correlation parameter ρ_v = 1/(a+b+1); (iii) a blind-spot atom of mass 1-π at α=1 caps the evidence extractable from any number of gates at -ln(1-π) nats, so reliability saturates below 1; (iv) letting the true-accept rate also vary (βsim H) yields a trichotomy -- gates eventually always help, plateau, or actively harm -- decided by the upper-tail exponents of G and H, with closed-form crossover k^dagger. The mechanism is survivorship: errors surviving gates are the high-α ones. The theory is measurable: R repeated verdicts per instance identify the first R moments of G, so two verdicts identify ρ_v; beta-binomial likelihood and NPMLE recover the reliability curve and the ill-posed ceiling. In synthetic tests, independence-based extrapolation underestimates failure by 20x at k=5 and ~3000x at k=10; the correlated fit at R=8 tracks held-out depths. The practical lever is decorrelation -- changing model family, modality, or evidence source -- not adding gates.

Using a large language model to play the user is now standard in scalable evaluation. It has a repeatedly diagnosed failure: the simulated user is excessively cooperative, so a system under test can score by the sheer number of questions it asks rather than by making the user willing to speak. We answer with a disclosure gate conditioning information release on the companion agent's behaviour: its state is a ladder of five ordered gates, merged onto three observable depth layers. We specify, ablate, and audit it, and train a user simulator against that specification. Gating behaviour is learned from the training corpus's synthetic branch, while the real branch supplies how people speak and react; after training, the simulator need not be told at runtime which gate each item sits behind. The gate is a load-bearing component of the environment: on the English corpus of a published companion-agent benchmark (CompanionBench), once training no longer states per example which gate each item sits behind, the largest rank displacement across 12 systems under test exceeds the noise band set by re-running that environment under a new seed, while per-system scores show no detectable change. We state two acceptance criteria: a ranking must be order-preserving, and absolute scores must be scale-stable. Of the candidates we examine, only one passes both -- the simulator we release -- and its leaderboard correlates at 0.993 with the benchmark's original simulator. By contrast, prompting a frontier model as the simulator barely moves the ranking while shifting every score upward -- a shift invisible to anyone checking the ranking alone. The environment we specify is the one that benchmark already used. That publication describes the mechanism in about four hundred words, and we supply what it lacked: specification, ablations, human studies, negative controls, and downstream sensitivity analysis.

Computers calculate transcendental functions by approximating them through the composition of a few limited-precision instructions. For example, an exponential can be calculated with a Taylor series. These approximation methods were developed over the centuries by mathematicians, who emphasized the attainability of arbitrary precision. Computers, however, operate on few limited precision types, such as the popular float32. In this study, we show that when aiming for limited precision, existing approximation methods can be outperformed by programs automatically discovered from scratch by a simple evolutionary algorithm. In particular, over real numbers, our method can approximate the exponential function reaching orders of magnitude more precision for a given number of operations when compared to previous approaches. More practically, over float32 numbers and constrained to less than 1 ULP of error, the same method attains a speedup over baselines by generating code that triggers better XLA/LLVM compilation paths. In other words, in both cases, evolution searched a vast space of possible programs, without knowledge of mathematics, to discover previously unknown optimized approximations to high precision, for the first time. We also give evidence that these results extend beyond the exponential. The ubiquity of transcendental functions suggests that our method has the potential to reduce the cost of scientific computing applications.

Test-time adaptation (TTA) typically assumes that model parameters can be updated at inference time. This assumption is restrictive for inference-only accelerators, frozen or third-party models, and memory-constrained deployments, and standard BatchNorm-based TTA configurations may also become inactive on architectures without BatchNorm. We study adaptation when the learned model must remain frozen. We introduce CASTER, a gradient-free method that stores source class statistics in a discriminative subspace, estimates a class-shared affine transformation from target-batch moments, and analytically transports the source class distributions before classification. CASTER requires no backward pass, optimizer state, or stored source feature bank. Across four backbones and seven datasets, it outperforms k-NN on identical frozen features in 27 of 28 backbone-dataset settings while retaining a median of 18x less state. Affine transport is not always reliable. On ImageNet-C, where batches contain only 64 samples for 1000 classes, unconditional transport loses 21.2 top-1 points. We therefore introduce an empirical residual-to-margin transportability certificate. Across 307 evaluation cells, every transport losing more than 10 points has certificate value above 3.9, although benign and destructive regimes are not perfectly separated. Gating converts an average -3.35-point effect of unconditional transport into a +1.69-point gain, and performance remains within 0.3 points of the best threshold over a broad threshold range. Finally, we show that this certificate is mechanism-specific: when applied to Tent, it accepts only 4.3% of updates and preserves 0.6% of Tent's available gain. These results position CASTER as a lightweight adaptation mechanism for frozen-model deployment, together with an explicit account of when its safety signal is informative and when it is not.

Dense-to-sparse gating mixture of experts (MoE) has recently become an effective alternative to a well-known sparse MoE. Rather than fixing the number of activated experts as in the latter model, which could limit the investigation of potential experts, the former model utilizes the temperature to control the softmax weight distribution and the sparsity of the MoE during training in order to stabilize the expert specialization. Nevertheless, while there are previous attempts to theoretically comprehend the sparse MoE, a comprehensive analysis of the dense-to-sparse gating MoE has remained elusive. Therefore, we aim to explore the impacts of the dense-to-sparse gate on the maximum likelihood estimation under the Gaussian MoE in this paper. We demonstrate that due to interactions between the temperature and other model parameters via some partial differential equations, the convergence rates of parameter estimations are slower than any polynomial rates, and could be as slow as O(1/log(n)), where n denotes the sample size. To address this issue, we propose using a novel activation dense-to-sparse gate, which routes the output of a linear layer to an activation function before delivering them to the softmax function. By imposing linearly independence conditions on the activation function and its derivatives, we show that the parameter estimation rates are significantly improved to polynomial rates.

Top-K sparse softmax gating mixture of experts has been widely used for scaling up massive deep-learning architectures without increasing the computational cost. Despite its popularity in real-world applications, the theoretical understanding of that gating function has remained an open problem. The main challenge comes from the structure of the top-K sparse softmax gating function, which partitions the input space into multiple regions with distinct behaviors. By focusing on a Gaussian mixture of experts, we establish theoretical results on the effects of the top-K sparse softmax gating function on both density and parameter estimations. Our results hinge upon defining novel loss functions among parameters to capture different behaviors of the input regions. When the true number of experts k_{ast} is known, we demonstrate that the convergence rates of density and parameter estimations are both parametric on the sample size. However, when k_{ast} becomes unknown and the true model is over-specified by a Gaussian mixture of k experts where k > k_{ast}, our findings suggest that the number of experts selected from the top-K sparse softmax gating function must exceed the total cardinality of a certain number of Voronoi cells associated with the true parameters to guarantee the convergence of the density estimation. Moreover, while the density estimation rate remains parametric under this setting, the parameter estimation rates become substantially slow due to an intrinsic interaction between the softmax gating and expert functions.

Linear Transformers have gained attention as efficient alternatives to standard Transformers, but their performance in retrieval and long-context tasks has been limited. To address these limitations, recent work has explored two distinct mechanisms: gating for adaptive memory control and the delta update rule for precise memory modifications. We observe that these mechanisms are complementary: gating enables rapid memory erasure while the delta rule facilitates targeted updates. Building on this insight, we introduce the gated delta rule and develop a parallel training algorithm optimized for modern hardware. Our proposed architecture, Gated DeltaNet, consistently surpasses existing models like Mamba2 and DeltaNet across multiple benchmarks, including language modeling, common-sense reasoning, in-context retrieval, length extrapolation, and long-context understanding. We further enhance performance by developing hybrid architectures that combine Gated DeltaNet layers with sliding window attention or Mamba2 layers, achieving both improved training efficiency and superior task performance.

Gated attention is an effective approach to mitigate attention sinks and enhance the representational capacity of attention. To further extend its effectiveness-efficiency Pareto frontier, we propose a Hybrid Gated Attention (HyGA) framework that contains three types of gating strategies. Specifically, these gates leverage diverse information from multiple stages of attention, and collaboratively build element-wise/head-wise gating from multiple perspectives, capturing intra-head and cross-head information interactions. Through our hybrid gating components, HyGA could provide multi-source modulation signals, enabling more comprehensive control over information flow and improving the representational capacity of attention. We also introduce low-rank matrix decomposition and learnable attention sink to further enhance training efficiency and stability. In experiments, we evaluate HyGA on widely-used benchmarks based on different backbones. The experimental results show that our HyGA comprehensively improves both training loss and various downstream performances compared with Gated attention. HyGA has also been verified to achieve the best performance at different computation costs, with comprehensive model analyses for better understanding. The proposed HyGA sheds light on a more effective, efficient, and stable attention mechanism.

Activation functions are core components of all deep learning architectures. Currently, the most popular activation functions are smooth ReLU variants like GELU and SiLU. These are self-gated activation functions where the range of the gating function is between zero and one. In this paper, we explore the viability of using arctan as a gating mechanism. A self-gated activation function that uses arctan as its gating function has a monotonically increasing first derivative. To make this activation function competitive, it is necessary to introduce a trainable parameter for every MLP block to expand the range of the gating function beyond zero and one. We find that this technique also improves existing self-gated activation functions. We conduct an empirical evaluation of Expanded ArcTan Linear Unit (xATLU), Expanded GELU (xGELU), and Expanded SiLU (xSiLU) and show that they outperform existing activation functions within a transformer architecture. Additionally, expanded gating ranges show promising results in improving first-order Gated Linear Units (GLU).

Large language models apply uniform computation to all inputs, regardless of difficulty. We propose PonderTTT, a gating strategy using the TTT layer's self-supervised reconstruction loss to selectively trigger Test-Time Training (TTT) updates. The gating decision itself is training-free--requiring no learned classifier or auxiliary networks; only a single scalar threshold is initially calibrated on unlabeled data and continuously adapted via EMA to maintain target update rates. Our experiments with GPT-2 models (124M to 1.5B) on code language modeling (The Stack v2, teacher-forced perplexity) demonstrate that this signal is inference-compatible, requiring no ground-truth labels. Our Reconstruction Gating achieves 82-89% Oracle Recovery while being fully training-free, significantly outperforming Random Skip baselines (up to 16% lower loss on OOD languages).

A single two-input gate suffices for all of Boolean logic in digital hardware. No comparable primitive has been known for continuous mathematics: computing elementary functions such as sin, cos, sqrt, and log has always required multiple distinct operations. Here I show that a single binary operator, eml(x,y)=exp(x)-ln(y), together with the constant 1, generates the standard repertoire of a scientific calculator. This includes constants such as e, pi, and i; arithmetic operations including addition, subtraction, multiplication, division, and exponentiation as well as the usual transcendental and algebraic functions. For example, exp(x)=eml(x,1), ln(x)=eml(1,eml(eml(1,x),1)), and likewise for all other operations. That such an operator exists was not anticipated; I found it by systematic exhaustive search and established constructively that it suffices for the concrete scientific-calculator basis. In EML (Exp-Minus-Log) form, every such expression becomes a binary tree of identical nodes, yielding a grammar as simple as S -> 1 | eml(S,S). This uniform structure also enables gradient-based symbolic regression: using EML trees as trainable circuits with standard optimizers (Adam), I demonstrate the feasibility of exact recovery of closed-form elementary functions from numerical data at shallow tree depths up to 4. The same architecture can fit arbitrary data, but when the generating law is elementary, it may recover the exact formula.

A great deal of effort has been devoted to reducing the risk of spurious scientific discoveries, from the use of sophisticated validation techniques, to deep statistical methods for controlling the false discovery rate in multiple hypothesis testing. However, there is a fundamental disconnect between the theoretical results and the practice of data analysis: the theory of statistical inference assumes a fixed collection of hypotheses to be tested, or learning algorithms to be applied, selected non-adaptively before the data are gathered, whereas in practice data is shared and reused with hypotheses and new analyses being generated on the basis of data exploration and the outcomes of previous analyses. In this work we initiate a principled study of how to guarantee the validity of statistical inference in adaptive data analysis. As an instance of this problem, we propose and investigate the question of estimating the expectations of m adaptively chosen functions on an unknown distribution given n random samples. We show that, surprisingly, there is a way to estimate an exponential in n number of expectations accurately even if the functions are chosen adaptively. This gives an exponential improvement over standard empirical estimators that are limited to a linear number of estimates. Our result follows from a general technique that counter-intuitively involves actively perturbing and coordinating the estimates, using techniques developed for privacy preservation. We give additional applications of this technique to our question.

Conditional computation is a popular strategy to make Transformers more efficient. Existing methods often target individual modules (e.g., mixture-of-experts layers) or skip layers independently of one another. However, interpretability research has demonstrated that the middle layers of Transformers exhibit greater redundancy, and that early layers aggregate information into token positions. Guided by these insights, we propose a novel architecture that dynamically skips a variable number of layers from the middle outward. In particular, a learned gating mechanism determines whether to bypass a symmetric span of central blocks based on the input, and a gated attention mechanism prevents subsequent tokens from attending to skipped token positions. Residual norms are controlled with a 'sandwich' or 'perilayernorm' scheme and gate sparsity with an adaptive regularization loss. We had aimed to reduce compute requirements for 'simpler' tokens and potentially foster an emergent multi-level representational hierarchy but, at the scales investigated, our approach does not achieve improvements in the trade-off between validation cross-entropy and estimated FLOPs compared to dense baselines with fewer layers. We release our code at https://github.com/tim-lawson/skip-middle.

Sequence models must decide what to write into memory and what to retain. In quantum and quantum-inspired sequence learning, nonlinear recurrent updates often require repeated circuit evaluations and sequential backpropagation through time, making long contexts costly. Gated fast-weight programmers (FWPs) based on quantum-inspired Kolmogorov-Arnold networks (QKANs) alleviate this bottleneck by storing context in time-varying fast parameters. However, their scalar gate applies one retention-write balance to every fast-state coordinate, forcing all parameters to share a memory timescale. We introduce Self-Modulating QKAN-based FWPs, which replace this broadcast gate with low-rank-generated element-wise modulation of the new-proposal branch, a bounded old-state branch, or both. We further propose Complementary Matrix Gating (CMG), which uses one sigmoid matrix gate to retain the old state and its complement to write the new proposal. CMG provides coordinate-wise memory control while preserving the bounded convex update and affine prefix-scan structure of scalar gating, at the modulation-head cost of a single-branch rule. We compare four self-modulating rules with scalar gating across four FWP architectures combining classical and QKAN-based slow and fast programmers. Across seven single-step forecasting benchmarks and five sequence lengths, CMG gives the most consistent improvements for architectures whose fast programmer incorporates a QKAN-based module. In direct multi-step forecasting of Jaynes-Cummings and transmon-resonator dynamics simulated with CUDA-Q Dynamics, CMG models maintain mean-squared errors on the order of 0.001 or lower across forecasting horizons of 4, 8, and 16 steps, while improving on their scalar-gated counterparts by at least 91.2%. These results establish coordinate-wise complementary modulation as a stable and effective update for QKAN-based FWPs.

Graphical user interface (GUI) grounding requires vision-language models (VLMs) to identify small target elements in high-resolution screenshots and predict precise screen coordinates. On-policy self-distillation (OPSD) is a promising post-training approach for this coordinate-sensitive task, since it provides dense token-level teacher signals beyond hard coordinate labels. However, naive OPSD is not well suited to GUI grounding: OPSD evaluates the teacher on student-generated prefixes, the quality of coordinate-token teacher signals can degrade when the prefix has already deviated from the target coordinate, leading to unreliable teacher signal. To mitigate this, We propose quality-aware self-distillation for VLM-based GUI grounding, which improves coordinate-token teacher-signal quality through soft correctness-aware gating and teacher-probability scaling. The soft correctness-aware gate checks whether the teacher's current coordinate-token prediction can still be completed into the ground-truth box under the student-generated prefix. If not, the corresponding teacher signal is down-weighted. Teacher-probability scaling then uses the teacher's confidence as a lightweight factor to further calibrate the strength of the gated supervision. A key empirical finding is that neither component alone improves overall performance, whereas combining them consistently improves performance. This suggests that the two mechanisms play complementary roles: correctness-aware gating suppresses unreliable coordinate-token supervision, while teacher-probability scaling calibrates the strength of the remaining signals. Experiments across six GUI grounding benchmarks show that our method consistently improves the base model and outperforms strong baselines.

In [7] Krotov and Hopfield suggest a generalized version of the well-known Hopfield model of associative memory. In their version they consider a polynomial interaction function and claim that this increases the storage capacity of the model. We prove this claim and take the "limit" as the degree of the polynomial becomes infinite, i.e. an exponential interaction function. With this interaction we prove that model has an exponential storage capacity in the number of neurons, yet the basins of attraction are almost as large as in the standard Hopfield model.

Temporal point processes (TPP) are a natural tool for modeling event-based data. Among all TPP models, Hawkes processes have proven to be the most widely used, mainly due to their adequate modeling for various applications, particularly when considering exponential or non-parametric kernels. Although non-parametric kernels are an option, such models require large datasets. While exponential kernels are more data efficient and relevant for specific applications where events immediately trigger more events, they are ill-suited for applications where latencies need to be estimated, such as in neuroscience. This work aims to offer an efficient solution to TPP inference using general parametric kernels with finite support. The developed solution consists of a fast ell_2 gradient-based solver leveraging a discretized version of the events. After theoretically supporting the use of discretization, the statistical and computational efficiency of the novel approach is demonstrated through various numerical experiments. Finally, the method's effectiveness is evaluated by modeling the occurrence of stimuli-induced patterns from brain signals recorded with magnetoencephalography (MEG). Given the use of general parametric kernels, results show that the proposed approach leads to an improved estimation of pattern latency than the state-of-the-art.

We study reinforcement learning for global decision-making in the presence of many local agents, where the global decision-maker makes decisions affecting all local agents, and the objective is to learn a policy that maximizes the rewards of both the global and the local agents. Such problems find many applications, e.g. demand response, EV charging, queueing, etc. In this setting, scalability has been a long-standing challenge due to the size of the state/action space which can be exponential in the number of agents. This work proposes the SUB-SAMPLE-Q algorithm where the global agent subsamples kleq n local agents to compute an optimal policy in time that is only exponential in k, providing an exponential speedup from standard methods that are exponential in n. We show that the learned policy converges to the optimal policy in the order of O(1/k+epsilon_{k,m}) as the number of sub-sampled agents k increases, where epsilon_{k,m} is the Bellman noise. We also conduct numerical simulations in a demand-response setting and a queueing setting.

Stochastic neurons and hard non-linearities can be useful for a number of reasons in deep learning models, but in many cases they pose a challenging problem: how to estimate the gradient of a loss function with respect to the input of such stochastic or non-smooth neurons? I.e., can we "back-propagate" through these stochastic neurons? We examine this question, existing approaches, and compare four families of solutions, applicable in different settings. One of them is the minimum variance unbiased gradient estimator for stochatic binary neurons (a special case of the REINFORCE algorithm). A second approach, introduced here, decomposes the operation of a binary stochastic neuron into a stochastic binary part and a smooth differentiable part, which approximates the expected effect of the pure stochatic binary neuron to first order. A third approach involves the injection of additive or multiplicative noise in a computational graph that is otherwise differentiable. A fourth approach heuristically copies the gradient with respect to the stochastic output directly as an estimator of the gradient with respect to the sigmoid argument (we call this the straight-through estimator). To explore a context where these estimators are useful, we consider a small-scale version of {\em conditional computation}, where sparse stochastic units form a distributed representation of gaters that can turn off in combinatorially many ways large chunks of the computation performed in the rest of the neural network. In this case, it is important that the gating units produce an actual 0 most of the time. The resulting sparsity can be potentially be exploited to greatly reduce the computational cost of large deep networks for which conditional computation would be useful.

Autoregressive Language Models (LLMs) trained on Next-Token Prediction (NTP) often suffer from ``Topic Drift'' where the generation wanders away from the initial prompt due to a reliance on local associations rather than global planning holtzman2019curious. While scaling model size mitigates this brown2020language, the fundamental myopia of the NTP objective remains. In this work, we introduce the Idea-Gated Transformer, a novel architecture that separates semantic planning from syntactic generation. We introduce an auxiliary ``Idea Head'' trained to predict the bag-of-words distribution for a future context window, creating a latent ``Concept Vector'' that actively gates the main vocabulary during generation. We propose a differentiable gating mechanism that suppresses semantically irrelevant tokens, effectively pruning the search space in real-time. Experiments on WikiText-103 demonstrate that while the Idea-Gated model achieves comparable validation perplexity to a standard GPT-2 baseline, it exhibits significantly superior Domain Retention. Qualitative and quantitative analysis reveals that the gating mechanism successfully locks generation into specific semantic clusters (e.g., Finance, Science) and resists associative drift, offering a parameter-efficient path toward more controllable language modeling.

Neural algorithmic reasoning is an emerging research direction that endows neural networks with the ability to mimic algorithmic executions step-by-step. A common paradigm in existing designs involves the use of historical embeddings in predicting the results of future execution steps. Our observation in this work is that such historical dependence intrinsically contradicts the Markov nature of algorithmic reasoning tasks. Based on this motivation, we present our ForgetNet, which does not use historical embeddings and thus is consistent with the Markov nature of the tasks. To address challenges in training ForgetNet at early stages, we further introduce G-ForgetNet, which uses a gating mechanism to allow for the selective integration of historical embeddings. Such an enhanced capability provides valuable computational pathways during the model's early training phase. Our extensive experiments, based on the CLRS-30 algorithmic reasoning benchmark, demonstrate that both ForgetNet and G-ForgetNet achieve better generalization capability than existing methods. Furthermore, we investigate the behavior of the gating mechanism, highlighting its degree of alignment with our intuitions and its effectiveness for robust performance.

Large language models, such as OpenAI's ChatGPT, have demonstrated exceptional language understanding capabilities in various NLP tasks. Sparsely activated mixture-of-experts (MoE) has emerged as a promising solution for scaling models while maintaining a constant number of computational operations. Existing MoE model adopts a fixed gating network where each token is computed by the same number of experts. However, this approach contradicts our intuition that the tokens in each sequence vary in terms of their linguistic complexity and, consequently, require different computational costs. Little is discussed in prior research on the trade-off between computation per token and model performance. This paper introduces adaptive gating in MoE, a flexible training strategy that allows tokens to be processed by a variable number of experts based on expert probability distribution. The proposed framework preserves sparsity while improving training efficiency. Additionally, curriculum learning is leveraged to further reduce training time. Extensive experiments on diverse NLP tasks show that adaptive gating reduces at most 22.5% training time while maintaining inference quality. Moreover, we conduct a comprehensive analysis of the routing decisions and present our insights when adaptive gating is used.

Timing and burst patterns can leak through encryption, and an adaptive adversary can exploit them. This undermines metadata-only detection in a stand-alone consumer gateway. Therefore, consumer gateways need streaming intrusion detection on encrypted traffic using metadata only, under tight CPU and latency budgets. We present a streaming IDS for stand-alone gateways that instantiates a lightweight two-state unit derived from Network-Optimised Spiking (NOS) dynamics per flow, named NOS-Gate. NOS-Gate scores fixed-length windows of metadata features and, under a K-of-M persistence rule, triggers a reversible mitigation that temporarily reduces the flow's weight under weighted fair queueing (WFQ). We evaluate NOS-Gate under timing-controlled evasion using an executable worlds benchmark that specifies benign device processes, auditable attacker budgets, contention structure, and packet-level WFQ replay to quantify queue impact. All methods are calibrated label-free via burn-in quantile thresholding. Across multiple reproducible worlds and malicious episodes, at an achieved 0.1% false-positive operating point, NOS-Gate attains 0.952 incident recall versus 0.857 for the best baseline in these runs. Under gating, it reduces p99.9 queueing delay and p99.9 collateral delay with a mean scoring cost of approx 2.09,μs per flow-window on CPU.

We provide a polynomial-time classical algorithm for noisy quantum circuits. The algorithm computes the expectation value of any observable for any circuit, with a small average error over input states drawn from an ensemble (e.g. the computational basis). Our approach is based upon the intuition that noise exponentially damps non-local correlations relative to local correlations. This enables one to classically simulate a noisy quantum circuit by only keeping track of the dynamics of local quantum information. Our algorithm also enables sampling from the output distribution of a circuit in quasi-polynomial time, so long as the distribution anti-concentrates. A number of practical implications are discussed, including a fundamental limit on the efficacy of noise mitigation strategies: for constant noise rates, any quantum circuit for which error mitigation is efficient on most input states, is also classically simulable on most input states.

The objective of continual learning (CL) is to learn tasks sequentially without retraining on earlier tasks. However, when subjected to CL, traditional neural networks exhibit catastrophic forgetting and limited generalization. To overcome these problems, we introduce a novel method called 'Gate and Obstruct Network' (GateON). GateON combines learnable gating of activity and online estimation of parameter relevance to safeguard crucial knowledge from being overwritten. Our method generates partially overlapping pathways between tasks which permits forward and backward transfer during sequential learning. GateON addresses the issue of network saturation after parameter fixation by a re-activation mechanism of fixed neurons, enabling large-scale continual learning. GateON is implemented on a wide range of networks (fully-connected, CNN, Transformers), has low computational complexity, effectively learns up to 100 MNIST learning tasks, and achieves top-tier results for pre-trained BERT in CL-based NLP tasks.

Efficient key-value (KV) cache management is crucial for the practical deployment of large language models (LLMs), yet existing compression techniques often incur a trade-off between performance degradation and computational overhead. We propose a novel gating-based KV cache eviction method for frozen-weight LLMs that achieves high compression ratios with negligible computational cost. Our approach introduces lightweight sink-attention gating modules to identify and retain critical KV pairs, and integrates seamlessly into both the prefill and decoding stages. The proposed gate training algorithm relies on forward passes of an LLM, avoiding expensive backpropagation, while achieving strong task generalization through a task-agnostic reconstruction objective. Extensive experiments across the Qwen2.5-1M, Qwen3, and Gemma3 families show that our method maintains near-lossless performance while evicting up to 70% of the KV cache. The results are consistent across a wide range of tasks, including long-context understanding, code comprehension, and mathematical reasoning, demonstrating the generality of our approach.

Zero-shot and in-context learning enable solving tasks without model fine-tuning, making them essential for developing generative model solutions. Therefore, it is crucial to understand whether a pretrained model can be prompted to approximate any function, i.e., whether it is a universal in-context approximator. While it was recently shown that transformer models do possess this property, these results rely on their attention mechanism. Hence, these findings do not apply to fully recurrent architectures like RNNs, LSTMs, and the increasingly popular SSMs. We demonstrate that RNNs, LSTMs, GRUs, Linear RNNs, and linear gated architectures such as Mamba and Hawk/Griffin can also serve as universal in-context approximators. To streamline our argument, we introduce a programming language called LSRL that compiles to these fully recurrent architectures. LSRL may be of independent interest for further studies of fully recurrent models, such as constructing interpretability benchmarks. We also study the role of multiplicative gating and observe that architectures incorporating such gating (e.g., LSTMs, GRUs, Hawk/Griffin) can implement certain operations more stably, making them more viable candidates for practical in-context universal approximation.

Understanding the parameter estimation of softmax gating Gaussian mixture of experts has remained a long-standing open problem in the literature. It is mainly due to three fundamental theoretical challenges associated with the softmax gating function: (i) the identifiability only up to the translation of parameters; (ii) the intrinsic interaction via partial differential equations between the softmax gating and the expert functions in the Gaussian density; (iii) the complex dependence between the numerator and denominator of the conditional density of softmax gating Gaussian mixture of experts. We resolve these challenges by proposing novel Voronoi loss functions among parameters and establishing the convergence rates of maximum likelihood estimator (MLE) for solving parameter estimation in these models. When the true number of experts is unknown and over-specified, our findings show a connection between the convergence rate of the MLE and a solvability problem of a system of polynomial equations.

Classical supply chain risk models treat node failures as statistically independent events, systematically underestimating cascade probabilities when supplier dependencies are strongly correlated. At n=40 nodes, the full correlated failure distribution requires O(2^n) classical samples, a regime where exact simulation demands 17.6 TB of memory and over 369,000 hours of computation on a standard workstation. We present QR-SPPS (Quantum-Native Retail Shock Propagation and Policy Stress Simulator), a three-algorithm quantum pipeline implemented using the Qiskit framework with the Aer statevector_simulator backend. First, a 40-node, 4-tier retail supply network is encoded as a 40-qubit Ising Hamiltonian using OpenFermion QubitOperator, where ZZ coupling terms encode correlated cascade probabilities structurally absent from classical Monte Carlo. Second, a hardware-efficient VQE circuit finds the ground-state stress distribution with zero error, detecting entangled cascade failures in 14/40 nodes with max|ΔP|=0.637 versus classical Monte Carlo. Third, we introduce the first application of ADAPT-VQE gradient screening to counterfactual macroeconomic policy evaluation: six crisis interventions are ranked in O(1) Qiskit operator evaluations per policy, a 287x speedup over sequential VQE re-optimisation. Fourth, Density-of-States QPE (DOS-QPE) reconstructs the full eigenspectrum via 32-step Trotter evolution and introduces a novel mapping of the Boltzmann catastrophe probability P_cat(T) to VIX-equivalent market volatility temperature, enabling direct integration into regulatory Value-at-Risk frameworks. Qiskit Aer scaling benchmarks confirm exponential classical intractability at 40 qubits.

Mixture-of-experts (MoE) model incorporates the power of multiple submodels via gating functions to achieve greater performance in numerous regression and classification applications. From a theoretical perspective, while there have been previous attempts to comprehend the behavior of that model under the regression settings through the convergence analysis of maximum likelihood estimation in the Gaussian MoE model, such analysis under the setting of a classification problem has remained missing in the literature. We close this gap by establishing the convergence rates of density estimation and parameter estimation in the softmax gating multinomial logistic MoE model. Notably, when part of the expert parameters vanish, these rates are shown to be slower than polynomial rates owing to an inherent interaction between the softmax gating and expert functions via partial differential equations. To address this issue, we propose using a novel class of modified softmax gating functions which transform the input value before delivering them to the gating functions. As a result, the previous interaction disappears and the parameter estimation rates are significantly improved.

Linear attention methods offer a compelling alternative to softmax attention due to their efficiency in recurrent decoding. Recent research has focused on enhancing standard linear attention by incorporating gating while retaining its computational benefits. Such Gated Linear Attention (GLA) architectures include competitive models such as Mamba and RWKV. In this work, we investigate the in-context learning capabilities of the GLA model and make the following contributions. We show that a multilayer GLA can implement a general class of Weighted Preconditioned Gradient Descent (WPGD) algorithms with data-dependent weights. These weights are induced by the gating mechanism and the input, enabling the model to control the contribution of individual tokens to prediction. To further understand the mechanics of this weighting, we introduce a novel data model with multitask prompts and characterize the optimization landscape of learning a WPGD algorithm. Under mild conditions, we establish the existence and uniqueness (up to scaling) of a global minimum, corresponding to a unique WPGD solution. Finally, we translate these findings to explore the optimization landscape of GLA and shed light on how gating facilitates context-aware learning and when it is provably better than vanilla linear attention.

Prior work has shown that fine-tuning large language models on malicious or incorrect outputs in narrow domains can induce broad misalignment and harmful behavior, a phenomenon known as emergent misalignment. However, efficient methods for reversing such misalignment remain limited. In this work, we make two contributions. First, we identify sycophancy fine-tuning, i.e., training models to passively agree with users' incorrect opinions, as a previously underexplored driver of emergent misalignment, and show that it induces broad and severe misaligned behavior. Second, we propose Alignment Gating, an efficient method for reversing emergent misalignment that inserts learnable and controllable gates into the model during fine-tuning. Through fine-tuning, these gates learn to identify the internal representations responsible for unsafe responses. Thus, amplifying or suppressing these representations then exacerbates or mitigates EM, respectively. We further find that alignment gating module exhibits strong generalization: gating weights obtained from narrow-domain fine-tuning substantially suppress broad-domain misaligned behavior while preserving the model's general capabilities.

Forecasting on sparse multivariate time series (MTS) aims to model the predictors of future values of time series given their incomplete past, which is important for many emerging applications. However, most existing methods process MTS's individually, and do not leverage the dynamic distributions underlying the MTS's, leading to sub-optimal results when the sparsity is high. To address this challenge, we propose a novel generative model, which tracks the transition of latent clusters, instead of isolated feature representations, to achieve robust modeling. It is characterized by a newly designed dynamic Gaussian mixture distribution, which captures the dynamics of clustering structures, and is used for emitting timeseries. The generative model is parameterized by neural networks. A structured inference network is also designed for enabling inductive analysis. A gating mechanism is further introduced to dynamically tune the Gaussian mixture distributions. Extensive experimental results on a variety of real-life datasets demonstrate the effectiveness of our method.

Spiking Neural Networks (SNNs) have garnered significant attention as a central paradigm in neuromorphic computing, owing to their energy efficiency and biological plausibility. However, training deep SNNs has critically depended on explicit normalization schemes, leading to a trade-off between performance and biological realism. To resolve this conflict, we propose a normalization-free learning framework that incorporates lateral inhibition inspired by cortical circuits. Our framework replaces the traditional feedforward SNN layer with distinct excitatory (E) and inhibitory (I) neuronal populations that capture the key features of the cortical E-I interaction. The E-I circuit dynamically regulates neuronal activity through subtractive and divisive inhibition, which respectively control the excitability and gain of neurons. To stabilize end-to-end training of the biologically constrained SNNs, we propose two key techniques: E-I Init and E-I Prop. E-I Init is a dynamic parameter initialization scheme that balances excitatory and inhibitory inputs while performing gain control. E-I Prop decouples the backpropagation of the circuit from the forward pass, regulating gradient flow. Experiments across multiple datasets and network architectures demonstrate that our framework enables stable training of deep normalization-free SNNs with biological realism, achieving competitive performance. Therefore, our work not only provides a solution to training deep SNNs but also serves as a computational platform for further exploring the functions of E-I interaction in large-scale cortical computation. Code is available at https://github.com/vwOvOwv/DeepEISNN.

Risk-averse reinforcement learning finds application in various high-stakes fields. Unlike classical reinforcement learning, which aims to maximize expected returns, risk-averse agents choose policies that minimize risk, occasionally sacrificing expected value. These preferences can be framed through utility theory. We focus on the specific case of the exponential utility function, where we can derive the Bellman equations and employ various reinforcement learning algorithms with few modifications. However, these methods suffer from numerical instability due to the need for exponent computation throughout the process. To address this, we introduce a numerically stable and mathematically sound loss function based on the Itakura-Saito divergence for learning state-value and action-value functions. We evaluate our proposed loss function against established alternatives, both theoretically and empirically. In the experimental section, we explore multiple financial scenarios, some with known analytical solutions, and show that our loss function outperforms the alternatives.

In this paper, we study quantum Ordered Binary Decision Diagrams(OBDD) model; it is a restricted version of read-once quantum branching programs, with respect to "width" complexity. It is known that the maximal gap between deterministic and quantum complexities is exponential. But there are few examples of functions with such a gap. We present a new technique ("reordering") for proving lower bounds and upper bounds for OBDD with an arbitrary order of input variables if we have similar bounds for the natural order. Using this transformation, we construct a total function REQ such that the deterministic OBDD complexity of it is at least 2^{Ω(n / log n)}, and the quantum OBDD complexity of it is at most O(n^2/log n). It is the biggest known gap for explicit functions not representable by OBDDs of a linear width. Another function(shifted equality function) allows us to obtain a gap 2^{Ω(n)} vs O(n^2). Moreover, we prove the bounded error quantum and probabilistic OBDD width hierarchies for complexity classes of Boolean functions. Additionally, using "reordering" method we extend a hierarchy for read-k-times Ordered Binary Decision Diagrams (k-OBDD) of polynomial width, for k = o(n / log^3 n). We prove a similar hierarchy for bounded error probabilistic k-OBDDs of polynomial, superpolynomial and subexponential width. The extended abstract of this work was presented on International Computer Science Symposium in Russia, CSR 2017, Kazan, Russia, June 8 -- 12, 2017

The input tokens to Vision Transformers carry little semantic meaning as they are defined as regular equal-sized patches of the input image, regardless of its content. However, processing uniform background areas of an image should not necessitate as much compute as dense, cluttered areas. To address this issue, we propose a dynamic mixed-scale tokenization scheme for ViT, MSViT. Our method introduces a conditional gating mechanism that selects the optimal token scale for every image region, such that the number of tokens is dynamically determined per input. The proposed gating module is lightweight, agnostic to the choice of transformer backbone, and trained within a few epochs (e.g., 20 epochs on ImageNet) with little training overhead. In addition, to enhance the conditional behavior of the gate during training, we introduce a novel generalization of the batch-shaping loss. We show that our gating module is able to learn meaningful semantics despite operating locally at the coarse patch-level. We validate MSViT on the tasks of classification and segmentation where it leads to improved accuracy-complexity trade-off.

Quantum technology has the potential to revolutionize how we acquire and process experimental data to learn about the physical world. An experimental setup that transduces data from a physical system to a stable quantum memory, and processes that data using a quantum computer, could have significant advantages over conventional experiments in which the physical system is measured and the outcomes are processed using a classical computer. We prove that, in various tasks, quantum machines can learn from exponentially fewer experiments than those required in conventional experiments. The exponential advantage holds in predicting properties of physical systems, performing quantum principal component analysis on noisy states, and learning approximate models of physical dynamics. In some tasks, the quantum processing needed to achieve the exponential advantage can be modest; for example, one can simultaneously learn about many noncommuting observables by processing only two copies of the system. Conducting experiments with up to 40 superconducting qubits and 1300 quantum gates, we demonstrate that a substantial quantum advantage can be realized using today's relatively noisy quantum processors. Our results highlight how quantum technology can enable powerful new strategies to learn about nature.

The Spiking Neural Network (SNN) has attracted more and more attention recently. It adopts binary spike signals to transmit information. Benefitting from the information passing paradigm of SNNs, the multiplications of activations and weights can be replaced by additions, which are more energy-efficient. However, its ``Hard Reset" mechanism for the firing activity would ignore the difference among membrane potentials when the membrane potential is above the firing threshold, causing information loss. Meanwhile, quantifying the membrane potential to 0/1 spikes at the firing instants will inevitably introduce the quantization error thus bringing about information loss too. To address these problems, we propose to use the ``Soft Reset" mechanism for the supervised training-based SNNs, which will drive the membrane potential to a dynamic reset potential according to its magnitude, and Membrane Potential Rectifier (MPR) to reduce the quantization error via redistributing the membrane potential to a range close to the spikes. Results show that the SNNs with the ``Soft Reset" mechanism and MPR outperform their vanilla counterparts on both static and dynamic datasets.

Energy-based models (EBMs) are known in the Machine Learning community for decades. Since the seminal works devoted to EBMs dating back to the noughties, there have been a lot of efficient methods which solve the generative modelling problem by means of energy potentials (unnormalized likelihood functions). In contrast, the realm of Optimal Transport (OT) and, in particular, neural OT solvers is much less explored and limited by few recent works (excluding WGAN-based approaches which utilize OT as a loss function and do not model OT maps themselves). In our work, we bridge the gap between EBMs and Entropy-regularized OT. We present a novel methodology which allows utilizing the recent developments and technical improvements of the former in order to enrich the latter. From the theoretical perspective, we prove generalization bounds for our technique. In practice, we validate its applicability in toy 2D and image domains. To showcase the scalability, we empower our method with a pre-trained StyleGAN and apply it to high-res AFHQ 512times 512 unpaired I2I translation. For simplicity, we choose simple short- and long-run EBMs as a backbone of our Energy-guided Entropic OT approach, leaving the application of more sophisticated EBMs for future research. Our code is available at: https://github.com/PetrMokrov/Energy-guided-Entropic-OT

Linear attention replaces the unbounded cache of softmax attention with a fixed-size recurrent state, reducing sequence mixing to linear time and decoding to constant memory. The hard part is not just what to forget, but how to edit this compressed memory without scrambling existing associations. Delta-rule models subtract the current read before writing a new value, and Kimi Delta Attention (KDA) sharpens forgetting with channel-wise decay. But the active edit still uses a single scalar gate to control two different things: how much old content to erase on the key side and how much new content to commit on the value side. We introduce Gated DeltaNet-2, which generalizes both Gated DeltaNet and KDA by inheriting adaptive forgetting and channel-wise decay while addressing their shared limitation, the scalar tie between erasing and writing. Gated Delta Rule-2 separates these roles with a channel-wise erase gate b_t and a channel-wise write gate w_t, reducing to KDA when both gates collapse to the same scalar and to Gated DeltaNet when the decay also collapses. We derive a fast-weight update view, a chunkwise WY algorithm with channel-wise decay absorbed into asymmetric erase factors, and a gate-aware backward pass that preserves efficient parallel training. At 1.3B parameters trained on 100B FineWeb-Edu tokens, Gated DeltaNet-2 achieves the strongest overall results among Mamba-2, Gated DeltaNet, KDA, and Mamba-3 variants across language modeling, commonsense reasoning, and retrieval. Its advantage is most pronounced on long-context RULER needle-in-a-haystack benchmarks, where it improves the evaluated multi-key retrieval setting and remains strong in both recurrent and hybrid settings. Code is available at https://github.com/NVlabs/GatedDeltaNet-2.

We introduce GateSkip, a simple residual-stream gating mechanism that enables token-wise layer skipping in decoder-only LMs. Each Attention/MLP branch is equipped with a sigmoid-linear gate that condenses the branch's output before it re-enters the residual stream. During inference we rank tokens by the gate values and skip low-importance ones using a per-layer budget. While early-exit or router-based Mixture-of-Depths models are known to be unstable and need extensive retraining, our smooth, differentiable gates fine-tune stably on top of pretrained models. On long-form reasoning, we save up to 15\% compute while retaining over 90\% of baseline accuracy. On instruction-tuned models we see accuracy gains at full compute and match baseline quality near 50\% savings. The learned gates give insight into transformer information flow (e.g., BOS tokens act as anchors), and the method combines easily with quantization, pruning, and self-speculative decoding.

Safety heads are lightweight classifiers attached to pretrained language models for flagging harmful inputs before generation. Their empirical detection performance has been studied, but their formal robustness properties remain largely unexplored. We ask when a State Space Model (SSM)-based safety head can be certified to produce the same prediction for all inputs within a bounded embedding-space perturbation. We prove that the answer turns on a single condition: the l_infty norm of the state transition matrix must satisfy A_infty<1 (the contraction condition), which enables exact interval bound propagation (IBP) certification for linear time-invariant classifiers. When the contraction condition holds, the reachable output interval has bounded steady-state width and examples can be certified as robustly classified. When it fails, the interval grows exponentially with sequence length and certification is impossible at any practical perturbation radius. We enforce contraction with a hinge penalty and show on toxic comment data that certified fraction improves from 41\% to 59\%, with a sharp empirical phase transition at A_infty=1 matching the theory. Applying a contraction-regularized S4 head to jailbreak detection on JailbreakBench, we achieve a zero-shot transfer to AdvBench (DR=0.994) and HarmBench (DR=0.988). A logistic regression on mean-pooled Mamba-130M embeddings matches or exceeds the S4 head on every detection metric, confirming that harmful intent is already linearly separable in the embedding space. The S4 safety head's contribution is not superior discrimination but the formal certification that no probe-based approach provides.

The performance of quantum annealing for combinatorial optimization is fundamentally limited by the minimum energy gap Δ encountered at quantum phase transitions. We investigate the scaling of Δ with system size N for two paradigmatic quantum spin-glass models: the two-dimensional Edwards-Anderson (2D-EA) and the all-to-all Sherrington-Kirkpatrick (SK) models. Utilizing a newly proposed unbiased energy-gap estimator for continuous-time projection quantum Monte Carlo simulations, complemented by high-performance sparse eigenvalue solvers, we characterize the gap distributions across disorder realizations. It is found that, in the 2D-EA case, the inverse-gap distribution develops a fat tail with infinite variance as N increases. This indicates that the unfavorable super-algebraic scaling of Δ, recently reported for binary couplings [Nature 631, 749 (2024)], persists for the Gaussian disorder considered here, pointing to a universal feature of 2D spin glasses. Conversely, the SK model retains a finite-variance distribution, with the disorder-averaged gap following a rather slow power law, close to Δpropto N^{-1/3}. This finding provides a promising outlook for the potential efficiency of quantum annealers for optimization problems with dense connectivity.

Time series forecasting (TSF) is critical in domains like energy, finance, healthcare, and logistics, requiring models that generalize across diverse datasets. Large pre-trained models such as Chronos and Time-MoE show strong zero-shot (ZS) performance but suffer from high computational costs. In this work, We introduce Super-Linear, a lightweight and scalable mixture-of-experts (MoE) model for general forecasting. It replaces deep architectures with simple frequency-specialized linear experts, trained on resampled data across multiple frequency regimes. A lightweight spectral gating mechanism dynamically selects relevant experts, enabling efficient, accurate forecasting. Despite its simplicity, Super-Linear matches state-of-the-art performance while offering superior efficiency, robustness to various sampling rates, and enhanced interpretability. The implementation of Super-Linear is available at https://github.com/azencot-group/SuperLinear{https://github.com/azencot-group/SuperLinear}

Quantum low-density parity-check (qLDPC) codes have a canonical mixed state---the normalized projector onto the code space---that lies at the intersection of quantum error correction and quantum many-body physics, yet the stability of its information-theoretic structure under local noise remains poorly understood. Recent progress on mixed-state phases of matter has identified the spatial decay of conditional mutual information (CMI) I(A:C|B) as a key diagnostic of such stability, analogous to the role of the spectral gap for quantum ground states. We prove that, for the normalized code-projector state of a Z_d qLDPC stabilizer code under local Pauli noise, the CMI across a buffer B separating A from C decays exponentially with the width of B in two complementary noise regimes independent of system size. At weak noise, this holds for full-system tripartitions, where correlations between the two sides require an error cluster to cross the buffer, an exponentially unlikely event. At strong noise, exponential decay holds whenever the buffer supports no logical operator, since correlations can persist only along rare chains of buffer qudits that remain informative. Under an additional space-uniformity assumption, we establish the robustness of Levin--Wen topological entropy. On finite-dimensional lattices of size n, the weak-noise bound further implies a constant-layer quasi-local Petz recovery channel of range O(poly(log n)/varepsilon) that recovers the noiseless state within trace-distance error varepsilon. Both CMI bounds apply to general qLDPC interaction graphs, including geometries where such constant-depth quasi-local recovery need not exist. Thus, the exponential decay of the noise-induced CMI provides a general criterion for mixed-state phase stability.

Recent advancements in Large Language Models (LLMs) have set themselves apart with their exceptional performance in complex language modelling tasks. However, these models are also known for their significant computational and storage requirements, primarily due to the quadratic computation complexity of softmax attention. To mitigate this issue, linear attention has been designed to reduce the quadratic space-time complexity that is inherent in standard transformers. In this work, we embarked on a comprehensive exploration of three key components that substantially impact the performance of the Gated Linear Attention module: feature maps, normalization, and the gating mechanism. We developed a feature mapping function to address some crucial issues that previous suggestions overlooked. Then we offered further rationale for the integration of normalization layers to stabilize the training process. Moreover, we explored the saturation phenomenon of the gating mechanism and augmented it with a refining module. We conducted extensive experiments and showed our architecture outperforms previous Gated Linear Attention mechanisms in extensive tasks including training from scratch and post-linearization with continual pre-training.

Adaptive prompting mechanisms have been proposed to enhance vision-language models by dynamically tailoring prompts to inputs. However, in frozen few-shot prompt learning with CLIP-style backbones, we systematically observe that adaptive gates and prompt-selection modules often collapse: they produce nearly constant outputs, contribute negligible gradient signals, and frequently fail to outperform fixed prompts. To further explore this issue, we present a systematic diagnostic study to uncover the underlying causes and conditions of adaptation failure. Through controlled experiments across datasets and multiple prompt learning architectures, we identify two recurring failure modes: gradient magnitude imbalance and gate degradation. Our findings invite a re-examination of indiscriminately adding architectural complexity in parameter-efficient learning and clarify when prompt-level adaptive gating is, and is not, effective in this regime.

Given the exponential growth of the volume of the ball w.r.t. its radius, the hyperbolic space is capable of embedding trees with arbitrarily small distortion and hence has received wide attention for representing hierarchical datasets. However, this exponential growth property comes at a price of numerical instability such that training hyperbolic learning models will sometimes lead to catastrophic NaN problems, encountering unrepresentable values in floating point arithmetic. In this work, we carefully analyze the limitation of two popular models for the hyperbolic space, namely, the Poincar\'e ball and the Lorentz model. We first show that, under the 64 bit arithmetic system, the Poincar\'e ball has a relatively larger capacity than the Lorentz model for correctly representing points. Then, we theoretically validate the superiority of the Lorentz model over the Poincar\'e ball from the perspective of optimization. Given the numerical limitations of both models, we identify one Euclidean parametrization of the hyperbolic space which can alleviate these limitations. We further extend this Euclidean parametrization to hyperbolic hyperplanes and exhibits its ability in improving the performance of hyperbolic SVM.

We present and experimentally realize a quantum algorithm for efficiently solving the following problem: given an Ntimes N matrix M, an N-dimensional vector emph{b}, and an initial vector emph{x}(0), obtain a target vector emph{x}(t) as a function of time t according to the constraint demph{x}(t)/dt=Memph{x}(t)+emph{b}. We show that our algorithm exhibits an exponential speedup over its classical counterpart in certain circumstances. In addition, we demonstrate our quantum algorithm for a 4times4 linear differential equation using a 4-qubit nuclear magnetic resonance quantum information processor. Our algorithm provides a key technique for solving many important problems which rely on the solutions to linear differential equations.

Autoregressive models with continuous tokens form a promising paradigm for visual generation, especially for text-to-image (T2I) synthesis, but they suffer from high computational cost. We study how to design compute-efficient linear attention within this framework. Specifically, we conduct a systematic empirical analysis of scaling behavior with respect to parameter counts under different design choices, focusing on (1) normalization paradigms in linear attention (division-based vs. subtraction-based) and (2) depthwise convolution for locality augmentation. Our results show that although subtraction-based normalization is effective for image classification, division-based normalization scales better for linear generative transformers. In addition, incorporating convolution for locality modeling plays a crucial role in autoregressive generation, consistent with findings in diffusion models. We further extend gating mechanisms, commonly used in causal linear attention, to the bidirectional setting and propose a KV gate. By introducing data-independent learnable parameters to the key and value states, the KV gate assigns token-wise memory weights, enabling flexible memory management similar to forget gates in language models. Based on these findings, we present LINA, a simple and compute-efficient T2I model built entirely on linear attention, capable of generating high-fidelity 1024x1024 images from user instructions. LINA achieves competitive performance on both class-conditional and T2I benchmarks, obtaining 2.18 FID on ImageNet (about 1.4B parameters) and 0.74 on GenEval (about 1.5B parameters). A single linear attention module reduces FLOPs by about 61 percent compared to softmax attention. Code and models are available at: https://github.com/techmonsterwang/LINA.

We study community recovery in the sparse symmetric stochastic block model with q communities, average degree d and signal strength λ through statistical decision theory and Fisher information, and obtain three characterizations of the Kesten-Stigum threshold dλ^2=1 and of the information-computation gap below it. First, on each community-size profile the minimax risk of any class of rules closed under averaging and vertex relabeling equals its Bayes risk under the uniform prior; the posterior mean is the unique Bayes rule and is admissible, and the Bayes risk of degree-D polynomial rules is the trivial risk times 1-Corr_D^2. Combined with known low-degree and information-theoretic results, this gives the gap as a worst-case statement: for qge 5 there is a window below the threshold in which no low-degree rule beats the trivial risk asymptotically, while an exponential-time rule does on a set of labelings of probability 1-o(1). Second, the Fisher information about λ carried by cycle counts is a series with terms of order k(dλ^2)^k, convergent exactly when dλ^2<1; below the threshold the relative error of every unbiased cycle-based estimator of λ^k stays above an explicit constant, and every cycle-count test has success probability bounded below one. Third, the derivative of belief propagation at its uninformative fixed point multiplies a random perturbation by |λ|d per iteration, and one EM step taken there leaves λ unchanged. A signal-to-noise computation recovers the condition dλ^{1/χ}>1 of Chin et al. for q=n^χ communities and identifies personalized PageRank as a walk count with suboptimal weights. Experiments on networks with up to 3times 10^5 vertices confirm the threshold for q=2, the hard window for q=5, and the many-community scaling.

Autoregressive sequence modeling stands as the cornerstone of modern Generative AI, powering results across diverse modalities ranging from text generation to image generation. However, a fundamental limitation of this paradigm is the rigid structural coupling of model capacity to computational cost: expanding a model's parametric memory -- its repository of factual knowledge or visual patterns -- traditionally requires deepening or widening the network, which incurs a proportional rise in active FLOPs. In this work, we introduce MoVE (Mixture of Value Embeddings), a mechanism that breaks this coupling and establishes a new axis for scaling capacity. MoVE decouples memory from compute by introducing a global bank of learnable value embeddings shared across all attention layers. For every step in the sequence, the model employs a differentiable soft gating mechanism to dynamically mix retrieved concepts from this bank into the standard value projection. This architecture allows parametric memory to be scaled independently of network depth by simply increasing the number of embedding slots. We validate MoVE through strictly controlled experiments on two representative applications of autoregressive modeling: Text Generation and Image Generation. In both domains, MoVE yields consistent performance improvements over standard and layer-wise memory baselines, enabling the construction of "memory-dense" models that achieve lower perplexity and higher fidelity than their dense counterparts at comparable compute budgets.

Quantum error correction provides a path to reach practical quantum computing by combining multiple physical qubits into a logical qubit, where the logical error rate is suppressed exponentially as more qubits are added. However, this exponential suppression only occurs if the physical error rate is below a critical threshold. In this work, we present two surface code memories operating below this threshold: a distance-7 code and a distance-5 code integrated with a real-time decoder. The logical error rate of our larger quantum memory is suppressed by a factor of Λ = 2.14 pm 0.02 when increasing the code distance by two, culminating in a 101-qubit distance-7 code with 0.143% pm 0.003% error per cycle of error correction. This logical memory is also beyond break-even, exceeding its best physical qubit's lifetime by a factor of 2.4 pm 0.3. We maintain below-threshold performance when decoding in real time, achieving an average decoder latency of 63 μs at distance-5 up to a million cycles, with a cycle time of 1.1 μs. To probe the limits of our error-correction performance, we run repetition codes up to distance-29 and find that logical performance is limited by rare correlated error events occurring approximately once every hour, or 3 times 10^9 cycles. Our results present device performance that, if scaled, could realize the operational requirements of large scale fault-tolerant quantum algorithms.

SwiGLU has become a standard gated activation in modern Transformer MLPs, yet its gate sharpness -- the smoothness and selectivity of the gating function -- is typically fixed throughout training. In this work, we propose Confidence-Aware SwiGLU (κ-SwiGLU), a variant of SwiGLU for Mixture-of-Experts (MoE) models that adjusts expert gate sharpness according to token-level routing confidence. Specifically, κ-SwiGLU parameterizes the SiLU gate sharpness coefficient as a learnable function of the router logit, enabling each expert gate unit to interpolate between smooth, broadly active gating and sharp, selective gating. We evaluate κ-SwiGLU on the FineWeb-Edu dataset across MoE Transformer models ranging from 8 to 28 layers. Across these settings, κ-SwiGLU improves mean CORE performance while adding negligible parameters and incurring only a small computational overhead, demonstrating that confidence-aware gate sharpness is a promising mechanism for improving MoE MLPs. The code is available at https://github.com/askerlee/kappa-swiglu.

Variational Autoencoders (VAEs) frequently suffer from posterior collapse, a failure mode in which the approximate posterior converges to the prior, rendering the latent code uninformative. Despite extensive research, a unified account of why collapse occurs has remained an open question. We identify and formalize two logically independent but coupled causes. Gradient imbalance occurs when the decoder's reconstruction signal vanishes faster than the KL regularization pressure as the posterior widens. Information gap occurs when the stochastic sampling step discards a substantial fraction of the encoder's computed representation, attenuating decoder sensitivity and making collapse inexpensive. Both causes share the same collapse trajectory, and we show that the information gap is algebraically equivalent to mismatch between the aggregate posterior and the prior, unifying two pathologies. Subsequently, we introduce λ-VAE, which resolves both causes through a single modification to the reparameterization step: the sampling noise is scaled by per-dimension exponent, while the KL penalty retains the original posterior variance. This asymmetry shifts the stable training attractor away from the degenerate collapsed state, driving all latent dimensions toward the same equilibrium -- a mechanism we term variance equalization. A closed-form optimal exponent per dimension follows from a net information gain objective, with a single hyperparameter controlling the reconstruction-generation tradeoff. We validate on standard benchmarks (Binary MNIST, Binary Omniglot, CIFAR-10, CelebA-64), showing consistent reductions in collapsed dimensions, information capacity gains of up to 2.8times nats, and reconstruction quality improvements of up to +0.33 BPD.

Gradient descent has proven to be a powerful and effective technique for optimization in numerous machine learning applications. Recent advances in computational neuroscience have shown that learning in standard gradient descent optimization formulation is not consistent with learning in biological systems. This has opened up interesting avenues for building biologically inspired learning techniques. One such approach is inspired by Dale's law, which states that inhibitory and excitatory synapses do not swap roles during the course of learning. The resulting exponential gradient descent optimization scheme leads to log-normally distributed synaptic weights. Interestingly, the density that satisfies the Fokker-Planck equation corresponding to the stochastic differential equation (SDE) with geometric Brownian motion (GBM) is the log-normal density. Leveraging this connection, we start with the SDE governing geometric Brownian motion, and show that discretizing the corresponding reverse-time SDE yields a multiplicative update rule, which surprisingly, coincides with the sampling equivalent of the exponential gradient descent update founded on Dale's law. Furthermore, we propose a new formalism for multiplicative denoising score-matching, subsuming the loss function proposed by Hyvaerinen for non-negative data. Indeed, log-normally distributed data is positive and the proposed score-matching formalism turns out to be a natural fit. This allows for training of score-based models for image data and results in a novel multiplicative update scheme for sample generation starting from a log-normal density. Experimental results on MNIST, Fashion MNIST, and Kuzushiji datasets demonstrate generative capability of the new scheme. To the best of our knowledge, this is the first instance of a biologically inspired generative model employing multiplicative updates, founded on geometric Brownian motion.

Differentiable matching layers and residual connection paradigms, often implemented via entropy-regularized Optimal Transport (OT), serve as critical mechanisms in structural prediction and architectural scaling. However, recovering discrete permutations or maintaining identity mappings via annealing εto 0 is notoriously unstable. In this work, we identify a fundamental mechanism for this failure: Premature Mode Collapse. By analyzing the non-normal dynamics of the Sinkhorn fixed-point map, we reveal a theoretical thermodynamic speed limit: standard exponential cooling outpaces the contraction rate of the inference operator, which degrades as O(1/ε). To address this, we propose Efficient Piecewise Hybrid Adaptive Stability Control (EPH-ASC), an adaptive scheduling algorithm that monitors the stability of the inference process. We demonstrate that EPH-ASC is essential for stabilizing Manifold-Constrained Hyper-Connections (mHC) during large-scale training on the FineWeb-Edu dataset, effectively preventing late-stage gradient explosions by enforcing a linear stability law.

The computational cost of transformer models makes them inefficient in low-latency or low-power applications. While techniques such as quantization or linear attention can reduce the computational load, they may incur a reduction in accuracy. In addition, globally reducing the cost for all inputs may be sub-optimal. We observe that for each layer, the full width of the layer may be needed only for a small subset of tokens inside a batch and that the "effective" width needed to process a token can vary from layer to layer. Motivated by this observation, we introduce the Adaptive Computation Module (ACM), a generic module that dynamically adapts its computational load to match the estimated difficulty of the input on a per-token basis. An ACM consists of a sequence of learners that progressively refine the output of their preceding counterparts. An additional gating mechanism determines the optimal number of learners to execute for each token. We also describe a distillation technique to replace any pre-trained model with an "ACMized" variant. The distillation phase is designed to be highly parallelizable across layers while being simple to plug-and-play into existing networks. Our evaluation of transformer models in computer vision and speech recognition demonstrates that substituting layers with ACMs significantly reduces inference costs without degrading the downstream accuracy for a wide interval of user-defined budgets.

Large models based on the Transformer architecture are susceptible to extreme-token phenomena, such as attention sinks and value-state drains. These issues, which degrade model performance, quantization fidelity, and interpretability, arise from a problematic mutual reinforcement mechanism where the model learns an inefficient 'no-op' behavior by focusing attention on tokens with near-zero value states. In this paper, we propose Value-State Gated Attention (VGA), a simple, dedicated, and stable architectural mechanism for performing 'no-op' attention efficiently by directly breaking this cycle. VGA introduces a learnable, data-dependent gate, computed directly from the value vectors (V), to modulate the output. Through a theoretical analysis of the underlying gradients, we show that gating the value-state with a function of itself is more effective at decoupling value and attention score updates than prior methods that gate on input embeddings. This creates a direct regulatory pathway that allows the model to suppress a token's contribution based on its emergent value representation. Our experiments demonstrate that VGA significantly mitigates the formation of attention sinks and stabilizes value-state norms, leading to improved performance, robust quantization fidelity, and enhanced model interpretability.

Training acceleration methods for diffusion transformers, such as REPA and dispersive regularization, are commonly explained by additional supervision, spatial-structure priors, or improved feature geometry. We revisit this view and uncover a striking internal change: the AdaLN-Zero residual gate α, which directly controls how conditioned updates enter the residual stream, is substantially reorganized by these methods. This observation motivates our hypothesis that acceleration arises, at least in part, from reshaping the residual gating pathway rather than solely from the feature structures explicitly imposed by auxiliary objectives. Supporting this view, accelerated models are markedly more sensitive to perturbations of α, and their gate geometry reflects semantic relationships among classes. We further introduce the normalized participation ratio (nPR) to characterize the channel-wise organization of the gates and find that, across REPA teachers, projector architectures, and radial-objective targets, broader gate participation accompanies better generation quality. Guided by this insight, we propose a simple teacher-free radial objective that promotes broader gate participation through feature-norm regularization, without pairwise feature comparisons. On ImageNet 256times256 with SiT-B/2, our method improves FID over dispersive loss while increasing mean gate participation over timesteps and classes, further supporting the link between gate organization and training acceleration. Our results identify residual-gate reorganization as a measurable axis of diffusion training acceleration and suggest a simple direction for further acceleration.

In physics and engineering literature, the distribution of the excursion-above-zero time distribution (exceedance distribution) for a stationary Gaussian process has been approximated by a stationary switching process with independently distributed switching times. The approach matched the covariance of the clipped Gaussian process with the one for the stationary switching process and the distribution of the latter was used as the so-called independent interval approximation (IIA). The approach successfully assessed the persistency exponent for many physically important processes but left an unanswered question when such an approach leads to a mathematically meaningful and proper exceedance distribution. Here we address this question by proposing an alternative matching of the expected values of the clipped Slepian process and the corresponding switched process initiated at the origin. The method has allowed resolving the mathematical correctness of the matching method for a large subclass of the Gaussian processes with monotonic covariance, for which we provide a sufficient condition for the validity of the IIA. Within this class, the IIA produces a valid distribution for the excursion time and is represented in an explicit stochastic form that connects directly to the covariance of the underlying Gaussian process. We compare the excursion level distributions as well as the corresponding persistency exponents obtained through the IIA method with numerically computed exact distributions, and the simulated distribution for several important Gaussian models. We also argue that for stationary Gaussian processes with a non-monotonic covariance, the IIA fails and should not be used.

In recent years, the application of transformer-based models in time-series forecasting has received significant attention. While often demonstrating promising results, the transformer architecture encounters challenges in fully exploiting the temporal relations within time series data due to its attention mechanism. In this work, we design eXponential Patch (xPatch for short), a novel dual-stream architecture that utilizes exponential decomposition. Inspired by the classical exponential smoothing approaches, xPatch introduces the innovative seasonal-trend exponential decomposition module. Additionally, we propose a dual-flow architecture that consists of an MLP-based linear stream and a CNN-based non-linear stream. This model investigates the benefits of employing patching and channel-independence techniques within a non-transformer model. Finally, we develop a robust arctangent loss function and a sigmoid learning rate adjustment scheme, which prevent overfitting and boost forecasting performance. The code is available at the following repository: https://github.com/stitsyuk/xPatch.

Auxiliary signal pathways in VLMs are routinely fitted with learnable gates so the optimiser can decide how much of the signal to admit. We find that the optimiser almost always decides on zero: across five injection designs, every gated pathway becomes behaviourally closed, with accuracy invariant to ablating the pathway at inference even when the gate parameter would nominally pass 30-45% of the signal. We attribute this suppression phenomenon to two regimes, a dead-gradient regime formalised through the caption-invariance of image-derived signals, and a negative-utility regime in which the auxiliary signal actively hurts the loss. Rather than fight suppression, we exploit it: we regularise LoRA fine-tuning with geometric auxiliary losses from hyperbolic visual relational graphs (IoA-driven entailment cones and angular repulsion on the Lorentz manifold), coupled only through the forward pass at training time and dropped at inference. Disaggregating GQA by question type exposes a clean dissociation. Three configurations without geometric losses at inference lose 2.85-3.39pp on relational questions while gaining ~1pp on attribute questions; a fourth that trains with the losses but infers through a soft prompt loses 5.14pp on rel for only +0.23pp on attr, so training-time regularisation alone does not protect relational accuracy without a geometric inference pathway. Configurations that keep the geometric pathway at inference preserve vanilla-level relational accuracy and match the attribute gain. Out of distribution on VSR, the RMS-prefix recipe preserves the spatial signal; stripping the geometric losses (G2) collapses VSR by 4.6pp, isolating them as the OOD source. A secondary result: embedding-norm alignment is necessary for generation-safe prefix injection, and learnable gates should be replaced with fixed, non-optional injection at matched scales.

LLM rerankers add of the order of \$0.2-0.3 per 1,000 queries and about a second of tail latency on top of a graph-augmented dense pipeline such as HippoRAG2, and on three multi-hop benchmarks they improve final-hop top-K coverage on seven of nine (dataset, K) cells, by up to +34.8 pp. We ask whether a learned per-query gate can skip the reranker where it will not help, using only features available before the LLM call (27 score and lexical statistics of the two retrieval lists plus a PCA of a small query embedding) with an executable fallback. Every choice, including the fallback and the threshold, is made inside the training fold and applied once to held-out queries, and harmful skips (the rerank would have found the target, the fallback did not) are reported next to the aggregate coverage. Across nine cells on 2WikiMultiHopQA, MuSiQue and HotpotQA the gate skips 51% of calls at an average held-out LastHop@K cost of 1.2 pp; four cells meet a pre-registered 1 pp rule, harmful skips occur in eight (190 harmful against 136 beneficial), and a random gate at the same skip rate loses 2 to 11 pp on the high-lift cells. A second rule sets each cell's threshold from a pre-specified budget on the expected harmful-skip rate over Platt-calibrated harm probabilities (ECE 0.025 after calibration, 0.094 before): at a 1 pp budget the gate skips 42% at -0.8 pp with 66 harmful skips and six cells within 1 pp, but realised harm exceeds the promise in six cells (mean 1.45 vs 0.83 pp), a selection optimism we quantify; a 0.5 pp budget realises about 1 pp. The harm probabilities are calibrated but barely discriminative (AUC 0.16 to 0.70). An earlier version reported 73% "lossless" savings; that figure rested on an oracle fallback and a wrong MuSiQue target, and we document both.

Massive activations are scalar values in transformer hidden states that achieve values orders of magnitude larger than typical activations and have been shown to be critical for model functionality. While prior work has characterized these phenomena in fully trained models, the temporal dynamics of their emergence during training remain poorly understood. We present the first comprehensive analysis of massive activation development throughout transformer training, using the Pythia model family as our testbed. Through systematic analysis of various model sizes across multiple training checkpoints, we demonstrate that massive activation emergence follows predictable mathematical patterns that can be accurately modeled using an exponentially-modulated logarithmic function with five key parameters. We develop a machine learning framework to predict these mathematical parameters from architectural specifications alone, achieving high accuracy for steady-state behavior and moderate accuracy for emergence timing and magnitude. These findings enable architects to predict and potentially control key aspects of massive activation emergence through design choices, with significant implications for model stability, training cycle length, interpretability, and optimization. Our findings demonstrate that the emergence of massive activations is governed by model design and can be anticipated, and potentially controlled, before training begins.

Latent variable models (LVMs) with discrete compositional latents are an important but challenging setting due to a combinatorially large number of possible configurations of the latents. A key tradeoff in modeling the posteriors over latents is between expressivity and tractable optimization. For algorithms based on expectation-maximization (EM), the E-step is often intractable without restrictive approximations to the posterior. We propose the use of GFlowNets, algorithms for sampling from an unnormalized density by learning a stochastic policy for sequential construction of samples, for this intractable E-step. By training GFlowNets to sample from the posterior over latents, we take advantage of their strengths as amortized variational inference algorithms for complex distributions over discrete structures. Our approach, GFlowNet-EM, enables the training of expressive LVMs with discrete compositional latents, as shown by experiments on non-context-free grammar induction and on images using discrete variational autoencoders (VAEs) without conditional independence enforced in the encoder.

Recurrent delta-rule models keep a fixed-size state matrix S (d_v x d_k) that compresses all past context. The state of the art (GDN-2) gates this update with element-wise matrix erase/write masks. This is powerful but has two defects. First, both gates are computed from the incoming token alone, making the model memory-blind: it decides what to erase without seeing what it has stored. Second, value-axis coupling in the erase gate blocks the WY-form triangular chunk solver that drives efficient training -- the intra-chunk system splits into d_v independent solves, collapsing throughput to serial-recurrence cost. We introduce CARVE (Content-Aware Recurrent with Value Efficiency), which fixes both and, via a single-launch "megakernel" scheduling of the same WY-form math, trains faster than the matrix-gated baseline it replaces. The key idea is architectural: restricting all gating to the key axis makes the intra-chunk coupling independent of the value index, restoring one unmodified WY-form solve. Within this constraint, CARVE conditions both gates on a content signal read once per chunk from the chunk-boundary state and folded algebraically into each gate's low-rank projection (by associativity, U(Sq)=(US)q), giving memory-aware gating at negligible extra traffic. At init the content projections are zero, so CARVE is bit-identical to the baseline; we prove the one-chunk staleness perturbs gates by only O(1/sqrt(L)), matching a measured 0.18% deviation flat up to L=128. At 1.3B parameters / 100B FineWeb-Edu tokens on H100 (three seeds), CARVE improves every axis: WikiText perplexity 15.72 vs 15.90 (hybrid 15.41 vs 15.62), +0.63 pp average common-sense accuracy, and state-of-the-art RULER and real-world recall -- while training +1.4% faster at matched depth and +19.3% at iso-quality depth, at +13% peak memory. Backed by six formal guarantees.

We present causal head gating (CHG), a scalable method for interpreting the functional roles of attention heads in transformer models. CHG learns soft gates over heads and assigns them a causal taxonomy - facilitating, interfering, or irrelevant - based on their impact on task performance. Unlike prior approaches in mechanistic interpretability, which are hypothesis-driven and require prompt templates or target labels, CHG applies directly to any dataset using standard next-token prediction. We evaluate CHG across multiple large language models (LLMs) in the Llama 3 model family and diverse tasks, including syntax, commonsense, and mathematical reasoning, and show that CHG scores yield causal, not merely correlational, insight validated via ablation and causal mediation analyses. We also introduce contrastive CHG, a variant that isolates sub-circuits for specific task components. Our findings reveal that LLMs contain multiple sparse task-sufficient sub-circuits, that individual head roles depend on interactions with others (low modularity), and that instruction following and in-context learning rely on separable mechanisms.

Continual Instruction Tuning (CIT) is adopted to continually instruct Large Models to follow human intent data by data. It is observed that existing gradient update would heavily destroy the performance on previous datasets during CIT process. Instead, Exponential Moving Average (EMA), owns the ability to trace previous parameters, which can aid in decreasing forgetting. Nonetheless, its stable balance weight fails to deal with the ever-changing datasets, leading to the out-of-balance between plasticity and stability. In this paper, we propose a general continual instruction tuning framework to address the challenge. Starting from the trade-off prerequisite and EMA update, we propose the plasticity and stability ideal condition. Based on Taylor expansion in the loss function, we find the optimal balance weight can be automatically determined by the gradients and learned parameters. Therefore, we propose a stable-plasticity balanced coefficient to avoid knowledge interference. Based on the semantic similarity of the instructions, we can determine whether to retrain or expand the training parameters and allocate the most suitable parameters for the testing instances. Extensive experiments across multiple continual instruction tuning benchmarks demonstrate that our approach not only enhances anti-forgetting capabilities but also significantly improves overall continual tuning performance. Our code is available at https://github.com/JingyangQiao/CoIN.

In a class of quantum circuits known as peaked circuits, the goal is to predict the most probable bit string at the output of the circuit. Since these circuits are designed to have a sharp peak in their output distribution, in principle it should be possible to simulate them using a truncated state vector with a limited number of terms, or a fraction of the total probability mass. This approximate simulation can be carried out on a classical computer with a sparse representation that stores only the nonzero amplitudes of the state vector, in contrast to the dense representations that are common in most quantum simulators. For efficiency, all operations on the state vector should be vectorized to the furthest possible extent and, if available, hardware acceleration can also be used. This work describes how these requirements were met in an open-source implementation, and discusses its performance and limitations.

We derive rates of exponential relative-entropy decay for a broad class of infinite-dimensional quantum Gibbs samplers. A key technical ingredient is a perturbative stability theory for hypercontractive estimates and logarithmic Sobolev inequalities. For suitably chosen dissipative couplings, we obtain exponential entropy decay for full multimode Bose-Hubbard samplers with arbitrary hopping and repulsive density interactions. For harmonically confined quantum Coulomb gases and molecular systems, we prove exponential entropy decay for every fixed finite-rank low-energy truncation of the interaction. Finally, we exhibit a periodic charging model with a positive spectral gap but no finite-time L^2-to-L^p bound for any p>2.

We expand the exponential kernel of attention into a truncated Taylor series of n terms and factorize each term into a tensor-power inner product, which collapses the causal history into exactly n recurrent states, each updated by one rank-one outer product per token and independent of context length N. The recursion is lossless with respect to the truncated kernel (2.2 times 10^{-16} in float64), and the states are additive across chunks (5.7 times 10^{-14}), so prefill admits a parallel scan. Measured exponents are t propto N^{2.28} for softmax against N^{0.91} for states, with a crossover at N approx 728 and 20.8times at N = 8192; per-token decoding latency is constant at 7.3,μs against N^{1.04} for softmax, giving 76.0times at N = 65536. Accuracy improves exponentially in n while cost grows combinatorially, since the m-th state is d+m-1{m} wide---a bound confirmed by SVD---so the optimum is second order, n = 3, at 6.5 times 10^{-4} error and 9{,}945 elements per head. Query and key normalization is a prerequisite: without it the argument exceeds the convergence radius and adding terms stops helping.

Time Series Forecasting (TSF) is highly vulnerable to backdoor attacks, yet effective defenses remain underexplored due to challenges arising from data entanglement and shifts in task formulation. To fill this gap, we conduct a systematic evaluation of thirteen representative backdoor defenses across the TSF life cycle and analyze their failure modes. Our results reveal two fundamental issues: (1) data entanglement induces channel-level signal dilution, rendering sample-filtering and trigger-synthesis defenses ineffective at localizing backdoors; and (2) task-formulation shift leads to training-loss degeneration, causing poisoned and clean windows to become indistinguishable at training stages. Based on these findings, we propose a training-time backdoor defense for TSF, termed TimeGuard. Our method adopts channel-wise pool training as the core paradigm and initializes a high-confidence pool using time-aware criteria to mitigate signal dilution. Moreover, we introduce distance-regularized loss selection to progressively expand the reliable pool during training and ease loss degeneration. Extensive experiments across multiple datasets, forecasting architectures, and TSF backdoor attacks demonstrate that TimeGuard substantially improves robustness, boosting MAE_P by 1.96times over the leading baseline, while preserving clean performance within 5% MAE_C.

We develop a discipline-agnostic emergence calculus that treats theories as fixed points of idempotent operators acting on descriptions. We show that, once processes are composable but access to the underlying system is mediated by a bounded observational interface, a canonical toolkit of six closure-changing primitives (P1--P6) is unavoidable. The framework unifies order-theoretic closure operators with dynamics-induced endomaps E_{τ,f} built from a Markov kernel, a coarse-graining lens, and a time scale τ. We introduce a computable total-variation idempotence defect for E_{τ,f}; small retention error implies approximate idempotence and yields stable "objects" packaged at the chosen τ within a fixed lens. For directionality, we define an arrow-of-time functional as the path-space KL divergence between forward and time-reversed trajectories and prove it is monotone under coarse-graining (data processing); we also formalize a protocol-trap audit showing that protocol holonomy alone cannot sustain asymmetry without a genuine affinity in the lifted dynamics. Finally, we prove a finite forcing-style counting lemma: relative to a partition-based theory, definable predicate extensions are exponentially rare, giving a clean anti-saturation mechanism for strict ladder climbing.

Mixture-of-Experts (MoE) models have gained popularity in achieving state-of-the-art performance in a wide range of tasks in computer vision and natural language processing. They effectively expand the model capacity while incurring a minimal increase in computation cost during training. However, deploying such models for inference is difficult due to their large size and complex communication pattern. In this work, we provide a characterization of two MoE workloads, namely Language Modeling (LM) and Machine Translation (MT) and identify their sources of inefficiencies at deployment. We propose three optimization techniques to mitigate sources of inefficiencies, namely (1) Dynamic gating, (2) Expert Buffering, and (3) Expert load balancing. We show that dynamic gating improves maximum throughput by 6.21-11.23times for LM, 5.75-10.98times for MT Encoder and 2.58-5.71times for MT Decoder. It also reduces memory usage by up to 1.36times for LM and up to 1.1times for MT. We further propose Expert Buffering, a new caching mechanism that only keeps hot, active experts in GPU memory while buffering the rest in CPU memory. This reduces static memory allocation by up to 1.47times. We finally propose a load balancing methodology that provides additional scalability to the workload.

Spiking Neural Networks (SNNs) have emerged as an energy-efficient alternative to conventional neural networks, demonstrating strong performance in computer vision and robotics. More recently, SNNs have been applied to time series forecasting (TSF), with methods exploring spiking temporal backbones, spike-compatible positional encodings, Fourier-domain processing, and redesigned neuron dynamics. However, existing SNN forecasting approaches process variables independently, lacking explicit mechanisms for modeling inter-variable dependencies. This is a critical limitation in multivariate settings, where cross-variable correlations carry substantial predictive information. We propose Spiking Fourier Graph Operators (SpikF-GO), which addresses this gap by combining a hypervariate graph formulation in which every scalar observation becomes a graph node with spike-driven spectral processing. SpikF-GO introduces a Hard Concrete frequency gate for learnable sparse frequency selection and a Complex LIF gate that applies independent spiking neurons to real and imaginary Fourier components, preserving binary, event-driven computation throughout the spectral domain. We further present a variant incorporating Central Pattern Generator-based positional encodings for stronger long-range temporal modeling. Evaluated on eight benchmarks under a unified experimental protocol, SpikF-GO achieves the best average rank among all SNN methods and outperforms its ANN counterpart, FourierGNN, at reduced energy cost. SpikF-GO maintains competitive accuracy even at substantially smaller embedding dimensions, thereby achieving significant energy reductions. To our knowledge, this is among the first works to bring graph-based multivariate modeling into the spiking domain for TSF and the first to provide a unified comparison across SNN forecasting architectures under a common experimental protocol.

Multi-controlled single-target (MC) gates are some of the most crucial building blocks for varied quantum algorithms. How to implement them optimally is thus a pivotal question. To answer this question in an architecture-independent manner, and to get a worst-case estimate, we should look at a linear nearest-neighbor (LNN) architecture, as this can be embedded in almost any qubit connectivity. Motivated by the above, here we describe a method which implements MC gates using no more than sim 4k+8n CNOT gates -- up-to 60% reduction over state-of-the-art -- while allowing for complete flexibility to choose the locations of n controls, the target, and a dirty ancilla out of k qubits. More strikingly, in case k approx n, our upper bound is sim 12n -- the best known for unrestricted connectivity -- and if n = 1, our upper bound is sim 4k -- the best known for a single long-range CNOT gate over k qubits -- therefore, if our upper bound can be reduced, then the cost of one or both of these simpler versions of MC gates will be immediately reduced accordingly. In practice, our method provides circuits that tend to require fewer CNOT gates than our upper bound for almost any given instance of MC gates.

Machine learning architectures, including transformers and recurrent neural networks (RNNs) have revolutionized forecasting in applications ranging from text processing to extreme weather. Notably, advanced network architectures, tuned for applications such as natural language processing, are transferable to other tasks such as spatiotemporal forecasting tasks. However, there is a scarcity of ablation studies to illustrate the key components that enable this forecasting accuracy. The absence of such studies, although explainable due to the associated computational cost, intensifies the belief that these models ought to be considered as black boxes. In this work, we decompose the key architectural components of the most powerful neural architectures, namely gating and recurrence in RNNs, and attention mechanisms in transformers. Then, we synthesize and build novel hybrid architectures from the standard blocks, performing ablation studies to identify which mechanisms are effective for each task. The importance of considering these components as hyper-parameters that can augment the standard architectures is exhibited on various forecasting datasets, from the spatiotemporal chaotic dynamics of the multiscale Lorenz 96 system, the Kuramoto-Sivashinsky equation, as well as standard real world time-series benchmarks. A key finding is that neural gating and attention improves the performance of all standard RNNs in most tasks, while the addition of a notion of recurrence in transformers is detrimental. Furthermore, our study reveals that a novel, sparsely used, architecture which integrates Recurrent Highway Networks with neural gating and attention mechanisms, emerges as the best performing architecture in high-dimensional spatiotemporal forecasting of dynamical systems.

Few neural architectures lend themselves to provable learning with gradient based methods. One popular model is the single-index model, in which labels are produced by composing an unknown linear projection with a possibly unknown scalar link function. Learning this model with SGD is relatively well-understood, whereby the so-called information exponent of the link function governs a polynomial sample complexity rate. However, extending this analysis to deeper or more complicated architectures remains challenging. In this work, we consider single index learning in the setting of symmetric neural networks. Under analytic assumptions on the activation and maximum degree assumptions on the link function, we prove that gradient flow recovers the hidden planted direction, represented as a finitely supported vector in the feature space of power sum polynomials. We characterize a notion of information exponent adapted to our setting that controls the efficiency of learning.

This work studies training instabilities of behavior cloning with deep neural networks. We observe that minibatch SGD updates to the policy network during training result in sharp oscillations in long-horizon rewards, despite negligibly affecting the behavior cloning loss. We empirically disentangle the statistical and computational causes of these oscillations, and find them to stem from the chaotic propagation of minibatch SGD noise through unstable closed-loop dynamics. While SGD noise is benign in the single-step action prediction objective, it results in catastrophic error accumulation over long horizons, an effect we term gradient variance amplification (GVA). We show that many standard mitigation techniques do not alleviate GVA, but find an exponential moving average (EMA) of iterates to be surprisingly effective at doing so. We illustrate the generality of this phenomenon by showing the existence of GVA and its amelioration by EMA in both continuous control and autoregressive language generation. Finally, we provide theoretical vignettes that highlight the benefits of EMA in alleviating GVA and shed light on the extent to which classical convex models can help in understanding the benefits of iterate averaging in deep learning.

Understanding when learning is statistically possible yet computationally hard is a central challenge in high-dimensional statistics. In this work, we investigate this question in the context of single- and multi-index models, classes of functions widely studied as benchmarks to probe the ability of machine learning methods to discover features in high-dimensional data. Our main contribution is to show that a Noise Sensitivity Exponent (NSE) - a simple quantity determined by the activation function - governs the existence and magnitude of statistical-to-computational gaps within a broad regime of these models. We first establish that, in single-index models with large additive noise, the onset of a computational bottleneck is fully characterized by the NSE. We then demonstrate that the same exponent controls a statistical-computational gap in the specialization transition of large separable multi-index models, where individual components become learnable. Finally, in hierarchical multi-index models, we show that the NSE governs the optimal computational rate in which different directions are sequentially learned. Taken together, our results identify the NSE as a unifying property linking noise robustness, computational hardness, and feature specialization in high-dimensional learning.

We show that in a drive-response coupling framework extreme events are suppressed in the response system by the dominance of a single driving signal. We validate this approach across three distinct response network topologies, namely (i) a pair of coupled neurons, (ii) a monolayer network of N coupled neurons and (iii) a two-layer multiplex network each composed of FitzHugh-Nagumo neuronal units. The response networks inherently exhibit extreme events. Our results demonstrate that influencing just one neuron in the response network with an appropriately tuned driving signal is sufficient to control extreme events across all three configurations. In the two-neuron case, suppression of extreme events occurs due to the breaking of phase-locking between the driving neuron and the targeted response neuron. In the case of monolayer and multiplex networks, suppression of extreme events results from the disruption of protoevent frequency dynamics and a subsequent frequency decoupling of the driven neuron from the rest of the network. We also observe that when the size of the neurons in response network connected to the drive increases, the onset of control occurs earlier indicating a scaling advantage of the method.

Transformers with linear attention allow for efficient parallel training but can simultaneously be formulated as an RNN with 2D (matrix-valued) hidden states, thus enjoying linear (with respect to output length) inference complexity. Recent works such as RetNet (Sun et al., 2023) and TransNormerLLM (Qin et al., 2023a) observe that adding a global decay term to the additive RNN update rule greatly improves performance, sometimes outperforming standard Transformers with softmax attention when trained at scale. In this work we show that adding a data-dependent gating mechanism further improves performance. We derive a parallel form of this gated linear attention layer that enables efficient training. However, a straightforward, numerically stable implementation of this parallel form requires generalized matrix multiplications in log-space for numerical stability, and thus cannot take advantage of tensor cores on modern GPUs which are optimized for standard matrix multiplications. We develop a hardware-efficient version of the parallel form that can still make use of tensor cores through block-parallel computations over sequence chunks. Experiments on moderate-scale language modeling (340M-parameter models trained on 15B tokens, 1.3B-parameter models trained on 100B tokens) show that gated linear attention (GLA) Transformers perform competitively against a strong LLaMA-architecture Transformer baseline (Touvron et al., 2023) as well as Mamba (Gu & Dao, 2023), a recently introduced state-space model with a data-dependent state transition mechanism. For training speed, our Triton-based implementation performs comparably to CUDA-optimized FlashAttention-2 (Dao, 2023) under the regular 2048 training length setting, while outperforming FlashAttention-2 when training on longer sequences beyond 4096.

There are few known exponential speedups for quantum algorithms and these tend to fall into even fewer families. One speedup that has mostly resisted generalization is the use of quantum walks to traverse the welded-tree graph, due to Childs, Cleve, Deotto, Farhi, Gutmann, and Spielman. We show how to generalize this to a large class of hierarchical graphs in which the vertices are grouped into "supervertices" which are arranged according to a d-dimensional lattice. Supervertices can have different sizes, and edges between supervertices correspond to random connections between their constituent vertices. The hitting times of quantum walks on these graphs are related to the localization properties of zero modes in certain disordered tight binding Hamiltonians. The speedups range from superpolynomial to exponential, depending on the underlying dimension and the random graph model. We also provide concrete realizations of these hierarchical graphs, and introduce a general method for constructing graphs with efficient quantum traversal times using graph sparsification.

When fitting the learning data of an individual to algorithm-like learning models, the observations are so dependent and non-stationary that one may wonder what the classical Maximum Likelihood Estimator (MLE) could do, even if it is the usual tool applied to experimental cognition. Our objective in this work is to show that the estimation of the learning rate cannot be efficient if the learning rate is constant in the classical Exp3 (Exponential weights for Exploration and Exploitation) algorithm. Secondly, we show that if the learning rate decreases polynomially with the sample size, then the prediction error and in some cases the estimation error of the MLE satisfy bounds in probability that decrease at a polynomial rate.

Standard transformer attention computes pairwise similarity between queries and keys, treating all tokens as equally salient regardless of their intrinsic informational content. In turbulent fluid dynamics, coherent structures -- the energetically dominant, spatially organized patterns that persist amid background chaos -- carry a disproportionate fraction of total energy and govern all transport. We propose that tokens play an analogous role in transformer attention: informationally dense positions (morphological boundaries, syntactic heads, discourse markers) concentrate spectral energy and should attract proportionally more attention than background tokens (function words, repeated patterns, low-information filler). We propose Energy-Gated Attention (EGA): a simple modification that gates value aggregation by the spectral energy of key token embeddings, computed by a single learned linear projection that discovers the dominant spectral mode of the embedding field. On TinyShakespeare, EGA achieves +0.103 validation loss improvement with only 12,480 additional parameters (<0.26% overhead) and no measurable computational cost. The result is consistent on Penn Treebank (+0.101), demonstrating dataset independence. A systematic ablation across three wavelet families (fixed Morlet, Daubechies db2/db4, and a parametric Morlet) establishes that fixed structured bases are suboptimal -- the optimal energy direction is data-adaptive and non-sinusoidal -- while identifying learned wavelet packets as a promising open direction. The learned energy threshold converges to tau ~= 0.35 independently of initialization, corresponding to the fraction (~36%) of tokens carrying above-average spectral energy in English text, a stable linguistic property consistent with the fraction of content words in running English text.

Recurrent neural networks (RNNs) compress the historical context into a memory state of fixed size, thus allowing for constant-time inference. The memory state size is a crucial factor in their performance, as exemplified by the strong performance and resurgence of linear attention, which extends the vector-valued hidden states of ordinary RNNs to matrix-valued hidden states. Crucially, linear attention does so in a parameter-efficient way, in particular by using an outer product of the key and value vectors to write to the matrix-valued hidden state. We generalize this construction and propose triadic linear attention, which writes the triadic outer product of a key, a second key, and a value, into a third-order (i.e., 3D) tensor state, and reads from it by contracting both key axes with two queries. An E-dimensional second key thus yields an E-fold increase in state size while adding only two projections. Triadic linear attention is compatible with data-dependent forgetting, the delta rule, and chunkwise-parallel training. Applied to Gated DeltaNet and scalar-gated linear attention, triadic linear attention substantially improves long-context language modeling and recall, outperforming alternatives that enlarge the state.

Continuous monitoring of generative AI systems requires detecting distribution shifts in the unstructured data they ingest and generate. Existing monitoring methods typically control false alarms by requiring the discrepancy measure used to detect shifts to be *pre-specified*. However, for unstructured data, pre-specification creates a sensitivity--breadth trade-off: a narrowly defined measure is sensitive to the anticipated shift but may miss unexpected changes, whereas a broadly defined measure may detect a wider range of shifts but with reduced sensitivity. Although adaptively hypothesizing the shift direction can overcome this limitation, such data-dependent adaptation generally invalidates conventional false-alarm guarantees. We reconcile adaptivity with statistical validity through testing-by-betting, reframing sequential changepoint detection as a betting game in which the bettor continually updates its hypothesized shift direction, or witness function, as new observations arrive. We introduce *asymptotic e-detectors*, which approximate the pre-change distribution using a calibration set of size m, and prove that they asymptotically control the average run length to a false alarm as m increases. Empirically, calibration sets as small as m=30 provide effective control. We then develop adaptive monitoring strategies ranging from black-box bettors that learn from embeddings to interpretable bettors based on LLM-proposed concepts. Across three real-world datasets, including data from a clinical AI scribe deployment, adaptive monitors detect shifts faster than fixed monitors at the same false-alarm level. Further, when the true shifts are sparse and interpretable, LLM-concept monitors provide the fastest and most interpretable detection, while also guiding root-cause diagnosis for complex shifts.

We study an algorithm-independent, worst-case lower bound for the Gaussian process (GP) bandit problem in the frequentist setting, where the reward function is fixed and has a bounded norm in the known reproducing kernel Hilbert space (RKHS). Specifically, we focus on the squared exponential (SE) kernel, one of the most widely used kernel functions in GP bandits. One of the remaining open questions for this problem is the gap in the dimension-dependent logarithmic factors between upper and lower bounds. This paper partially resolves this open question under a hyperspherical input domain. We show that any algorithm suffers Ω(T (ln T)^{d (ln ln T)^{-d}}) cumulative regret, where T and d represent the total number of steps and the dimension of the hyperspherical domain, respectively. Regarding the simple regret, we show that any algorithm requires Ω(ε^{-2}(ln 1ε)^d (ln ln 1ε)^{-d}) time steps to find an ε-optimal point. We also provide the improved O((ln T)^{d+1}(ln ln T)^{-d}) upper bound on the maximum information gain for the SE kernel. Our results guarantee the optimality of the existing best algorithm up to dimension-independent logarithmic factors under a hyperspherical input domain.

We study whether an aperiodic hierarchy can provide a structural advantage for lossless compression over periodic alternatives. We show that Fibonacci quasicrystal tilings avoid the finite-depth collapse that affects periodic hierarchies: usable n-gram lookup positions remain non-zero at every level, while periodic tilings collapse after O(log p) levels for period p. This yields an aperiodic hierarchy advantage: dictionary reuse remains available across all scales instead of vanishing beyond a finite depth. Our analysis gives four main consequences. First, the Golden Compensation property shows that the exponential decay in the number of positions is exactly balanced by the exponential growth in phrase length, so potential coverage remains scale-invariant with asymptotic value Wvarphi/5. Second, using the Sturmian complexity law p(n)=n+1, we show that Fibonacci/Sturmian hierarchies maximize codebook coverage efficiency among binary aperiodic tilings. Third, under long-range dependence, the resulting hierarchy achieves lower coding entropy than comparable periodic hierarchies. Fourth, redundancy decays super-exponentially with depth, whereas periodic systems remain locked at the depth where collapse occurs. We validate these results with Quasicryth, a lossless text compressor built on a ten-level Fibonacci hierarchy with phrase lengths {2,3,5,8,13,21,34,55,89,144}. In controlled A/B experiments with identical codebooks, the aperiodic advantage over a Period-5 baseline grows from 36{,}243 B at 3 MB to 11{,}089{,}469 B at 1 GB, explained by the activation of deeper hierarchy levels. On enwik9, Quasicryth achieves 225{,}918{,}349 B (22.59%), with 20{,}735{,}733 B saved by the Fibonacci tiling relative to no tiling.

The capacity of a neural network to absorb information is limited by its number of parameters. Conditional computation, where parts of the network are active on a per-example basis, has been proposed in theory as a way of dramatically increasing model capacity without a proportional increase in computation. In practice, however, there are significant algorithmic and performance challenges. In this work, we address these challenges and finally realize the promise of conditional computation, achieving greater than 1000x improvements in model capacity with only minor losses in computational efficiency on modern GPU clusters. We introduce a Sparsely-Gated Mixture-of-Experts layer (MoE), consisting of up to thousands of feed-forward sub-networks. A trainable gating network determines a sparse combination of these experts to use for each example. We apply the MoE to the tasks of language modeling and machine translation, where model capacity is critical for absorbing the vast quantities of knowledge available in the training corpora. We present model architectures in which a MoE with up to 137 billion parameters is applied convolutionally between stacked LSTM layers. On large language modeling and machine translation benchmarks, these models achieve significantly better results than state-of-the-art at lower computational cost.

This paper introduces a continuous-time stochastic dynamical framework for understanding how large language models (LLMs) may self-amplify latent biases or toxicity through their own chain-of-thought reasoning. The model posits an instantaneous "severity" variable x(t) in [0,1] evolving under a stochastic differential equation (SDE) with a drift term μ(x) and diffusion σ(x). Crucially, such a process can be consistently analyzed via the Fokker--Planck approach if each incremental step behaves nearly Markovian in severity space. The analysis investigates critical phenomena, showing that certain parameter regimes create phase transitions from subcritical (self-correcting) to supercritical (runaway severity). The paper derives stationary distributions, first-passage times to harmful thresholds, and scaling laws near critical points. Finally, it highlights implications for agents and extended LLM reasoning models: in principle, these equations might serve as a basis for formal verification of whether a model remains stable or propagates bias over repeated inferences.

Spiking Neural Networks (SNNs) as one of the biology-inspired models have received much attention recently. It can significantly reduce energy consumption since they quantize the real-valued membrane potentials to 0/1 spikes to transmit information thus the multiplications of activations and weights can be replaced by additions when implemented on hardware. However, this quantization mechanism will inevitably introduce quantization error, thus causing catastrophic information loss. To address the quantization error problem, we propose a regularizing membrane potential loss (RMP-Loss) to adjust the distribution which is directly related to quantization error to a range close to the spikes. Our method is extremely simple to implement and straightforward to train an SNN. Furthermore, it is shown to consistently outperform previous state-of-the-art methods over different network architectures and datasets.

Proper weight initialization prior to training has historically been one of the key factors that helped kick off the deep learning revolution. Initialization is even more crucial in "reservoir computing", where the weights of a readout layer are learned linearly while the reservoir weights are fixed and largely determine the richness, stability and memory of the resulting dynamics. In the infinite-width limit it has been shown that meaningful initializations are those sitting at an effective critical point of the randomly initialized model. The phase transition is controlled by the weight variance g^2 and separates an ordered phase from a chaotic one where information progressively degrades. Here we derive a simple criterion to estimate the critical g_c for a broad class of recurrent architectures and we show that it closely tracks the gain at which a gated-RNN reservoir achieves peak performance on a chaotic forecasting task. Finally, we argue that our criterion can serve as a design principle for future initialization schemes.

Transformers have been actively studied for time-series forecasting in recent years. While often showing promising results in various scenarios, traditional Transformers are not designed to fully exploit the characteristics of time-series data and thus suffer some fundamental limitations, e.g., they generally lack of decomposition capability and interpretability, and are neither effective nor efficient for long-term forecasting. In this paper, we propose ETSFormer, a novel time-series Transformer architecture, which exploits the principle of exponential smoothing in improving Transformers for time-series forecasting. In particular, inspired by the classical exponential smoothing methods in time-series forecasting, we propose the novel exponential smoothing attention (ESA) and frequency attention (FA) to replace the self-attention mechanism in vanilla Transformers, thus improving both accuracy and efficiency. Based on these, we redesign the Transformer architecture with modular decomposition blocks such that it can learn to decompose the time-series data into interpretable time-series components such as level, growth and seasonality. Extensive experiments on various time-series benchmarks validate the efficacy and advantages of the proposed method. Code is available at https://github.com/salesforce/ETSformer.

Fine-tuning is the primary mechanism for adapting foundation models to downstream tasks; however, standard approaches largely optimize task objectives in isolation and do not account for secondary yet critical alignment objectives (e.g., safety and hallucination avoidance). As a result, downstream fine-tuning can degrade alignment and fail to correct pre-existing misaligned behavior. We propose an alignment-aware fine-tuning framework that integrates feedback from an external alignment signal through policy-gradient-based regularization. Our method introduces an adaptive gating mechanism that dynamically balances supervised and alignment-driven gradients on a per-sample basis, prioritizing uncertain or misaligned cases while allowing well-aligned examples to follow standard supervised updates. The framework further learns abstention behavior for fully misaligned inputs, incorporating conservative responses directly into the fine-tuned model. Experiments on general and domain-specific instruction-tuning benchmarks demonstrate consistent reductions in harmful and hallucinated outputs without sacrificing downstream task performance. Additional analyses show robustness to adversarial fine-tuning, prompt-based attacks, and unsafe initializations, establishing adaptively gated alignment optimization as an effective approach for alignment-preserving and alignment-recovering model adaptation.

The Transformer architecture, underpinned by the self-attention mechanism, has become the de facto standard for sequence modeling tasks. However, its core computational primitive scales quadratically with sequence length (O(N^2)), creating a significant bottleneck for processing long contexts. In this paper, we propose the Gated Associative Memory (GAM) network, a novel, fully parallel architecture for sequence modeling that exhibits linear complexity (O(N)) with respect to sequence length. The GAM block replaces the self-attention layer with two parallel pathways: a causal convolution to efficiently capture local, position-dependent context, and a parallel associative memory retrieval mechanism to model global, content-based patterns. These pathways are dynamically fused using a gating mechanism, allowing the model to flexibly combine local and global information for each token. We implement GAM from scratch and conduct a rigorous comparative analysis against a standard Transformer model and a modern linear-time baseline (Mamba) on the WikiText-2 benchmark, as well as against the Transformer on the TinyStories dataset. Our experiments demonstrate that GAM is consistently faster, outperforming both baselines on training speed, and achieves a superior or competitive final validation perplexity across all datasets, establishing it as a promising and efficient alternative for sequence modeling.

We study superspace concentration as a quantum resource, formalized through the focus measure F(ho) = λ_max(ho_super) - the largest eigenvalue of the reduced superspace state - which quantifies the capacity of a quantum system to concentrate informational weight into a preferred subspace of an extended degree-of-freedom space. We develop a complete resource-theoretic framework around this measure and validate its properties through GPU-accelerated numerical simulation. Analytic decoherence predictions are confirmed to machine precision (1.11 x 10^{-16}) for superspace dimensions dS in {2,4,8,16,32}. Focus monotonicity holds across 10,000 random states with zero violations under four focus-non-generating channels across six system configurations. Focused quantum states resist coherent unitary attacks with significantly greater resilience than standard fidelity predicts, with focus remaining above 0.9 at attack strength ε = 0.302 versus ε = 0.174 for fidelity. We further demonstrate that the focus measure and the U(dS)-asymmetry measure are operationally distinct: asymmetry remains near zero and provides no robustness signal under coherent and targeted attacks while focus tracks spectral concentration and remains robust until ε > 0.3. The connection between Grover's algorithm and superspace concentration is made explicit via the identity F(|ψ_k><ψ_k|) = P(marked), providing a resource-theoretic interpretation of oracle query complexity. Finally, we provide the first numerical characterization of the focus capacity gap ΔF, identifying a log_2(dS) scaling law confirmed for both product and correlated noise channels.

Multi-agent language systems can exhibit a failure mode where a shared dominant context progressively absorbs individual semantics, yielding near-uniform behavior across agents. We study this effect under the name Asymptotic Semantic Collapse in Hierarchical Optimization. In a closed linguistic setting with a Dominant Anchor Node whose semantic state has effectively infinite inertia, we show that repeated interactions with Peripheral Agent Nodes drive an asymptotic alignment that minimizes a global loss. We model semantic states as points on a Riemannian manifold and analyze the induced projection dynamics. Two consequences follow. First, the limiting semantic configuration is insensitive to the optimization history: both smooth gradient-style updates and stochastic noisy updates converge to the same topological endpoint, establishing path independence at convergence. Second, the degree of context dependence controls information content: moving from atomic (independent) representations to fully entangled (context-bound) representations forces the node entropy, interpreted as available degrees of freedom, to vanish in the limit. The theory connects information-theoretic quantities with differential-geometric structure and suggests an interpretation as an immutable consensus rule that constrains agents to a shared semantic grammar. A lightweight dataset-free benchmark on an RWKV-7 13B GGUF checkpoint complements the analysis, reporting zero hash collisions, mean compliance of 0.50 under greedy decoding and 0.531 under stochastic decoding, and final Jaccard-to-anchor similarity values of 0.295 and 0.224, respectively.

Diffusion models recently proved to be remarkable priors for Bayesian inverse problems. However, training these models typically requires access to large amounts of clean data, which could prove difficult in some settings. In this work, we present a novel method based on the expectation-maximization algorithm for training diffusion models from incomplete and noisy observations only. Unlike previous works, our method leads to proper diffusion models, which is crucial for downstream tasks. As part of our method, we propose and motivate an improved posterior sampling scheme for unconditional diffusion models. We present empirical evidence supporting the effectiveness of our method.

Kernel methods in Quantum Machine Learning (QML) have recently gained significant attention as a potential candidate for achieving a quantum advantage in data analysis. Among other attractive properties, when training a kernel-based model one is guaranteed to find the optimal model's parameters due to the convexity of the training landscape. However, this is based on the assumption that the quantum kernel can be efficiently obtained from quantum hardware. In this work we study the performance of quantum kernel models from the perspective of the resources needed to accurately estimate kernel values. We show that, under certain conditions, values of quantum kernels over different input data can be exponentially concentrated (in the number of qubits) towards some fixed value. Thus on training with a polynomial number of measurements, one ends up with a trivial model where the predictions on unseen inputs are independent of the input data. We identify four sources that can lead to concentration including: expressivity of data embedding, global measurements, entanglement and noise. For each source, an associated concentration bound of quantum kernels is analytically derived. Lastly, we show that when dealing with classical data, training a parametrized data embedding with a kernel alignment method is also susceptible to exponential concentration. Our results are verified through numerical simulations for several QML tasks. Altogether, we provide guidelines indicating that certain features should be avoided to ensure the efficient evaluation of quantum kernels and so the performance of quantum kernel methods.

In this paper, we study the flow of signals through linear paths with the nonlinear condition that a node emits a signal when it receives external stimuli or when two incoming signals from other nodes arrive coincidentally with a combined amplitude above a fixed threshold. Sets of such nodes form a polychrony group and can sometimes lead to cascades. In the context of this work, cascades are polychrony groups in which the number of nodes activated as a consequence of other nodes is greater than the number of externally activated nodes. The difference between these two numbers is the so-called profit. Given the initial conditions, we predict the conditions for a vertex to activate at a prescribed time and provide an algorithm to efficiently reconstruct a cascade. We develop a dictionary between polychrony groups and graph theory. We call the graph corresponding to a cascade a chinampa. This link leads to a topological classification of chinampas. We enumerate the chinampas of profits zero and one and the description of a family of chinampas isomorphic to a family of partially ordered sets, which implies that the enumeration problem of this family is equivalent to computing the Stanley-order polynomials of those partially ordered sets.

Spiking neural networks offer low energy consumption due to their event-driven nature. Beyond binary spike outputs, their intrinsic floating-point dynamics merit greater attention. Neuronal threshold levels and reset modes critically determine spike count and timing. Hard reset cause information loss, while soft reset apply uniform treatment to neurons. To address these issues, we design an adaptive reset neuron that establishes relationships between inputs, outputs, and reset, while integrating a simple yet effective threshold adjustment strategy. Experimental results demonstrate that our method achieves excellent performance while maintaining lower energy consumption. In particular, it attains state-of-the-art accuracy on Tiny-ImageNet and CIFAR10-DVS. Codes are available at https://github.com/2ephyrus/AR-LIF.

Paging is a prototypical problem in the area of online algorithms. It has also played a central role in the development of learning-augmented algorithms -- a recent line of research that aims to ameliorate the shortcomings of classical worst-case analysis by giving algorithms access to predictions. Such predictions can typically be generated using a machine learning approach, but they are inherently imperfect. Previous work on learning-augmented paging has investigated predictions on (i) when the current page will be requested again (reoccurrence predictions), (ii) the current state of the cache in an optimal algorithm (state predictions), (iii) all requests until the current page gets requested again, and (iv) the relative order in which pages are requested. We study learning-augmented paging from the new perspective of requiring the least possible amount of predicted information. More specifically, the predictions obtained alongside each page request are limited to one bit only. We consider two natural such setups: (i) discard predictions, in which the predicted bit denotes whether or not it is ``safe'' to evict this page, and (ii) phase predictions, where the bit denotes whether the current page will be requested in the next phase (for an appropriate partitioning of the input into phases). We develop algorithms for each of the two setups that satisfy all three desirable properties of learning-augmented algorithms -- that is, they are consistent, robust and smooth -- despite being limited to a one-bit prediction per request. We also present lower bounds establishing that our algorithms are essentially best possible.

While diffusion models can learn complex distributions, sampling requires a computationally expensive iterative process. Existing distillation methods enable efficient sampling, but have notable limitations, such as performance degradation with very few sampling steps, reliance on training data access, or mode-seeking optimization that may fail to capture the full distribution. We propose EM Distillation (EMD), a maximum likelihood-based approach that distills a diffusion model to a one-step generator model with minimal loss of perceptual quality. Our approach is derived through the lens of Expectation-Maximization (EM), where the generator parameters are updated using samples from the joint distribution of the diffusion teacher prior and inferred generator latents. We develop a reparametrized sampling scheme and a noise cancellation technique that together stabilizes the distillation process. We further reveal an interesting connection of our method with existing methods that minimize mode-seeking KL. EMD outperforms existing one-step generative methods in terms of FID scores on ImageNet-64 and ImageNet-128, and compares favorably with prior work on distilling text-to-image diffusion models.

Recent years have witnessed Spiking Neural Networks (SNNs) gaining attention for their ultra-low energy consumption and high biological plausibility compared with traditional Artificial Neural Networks (ANNs). Despite their distinguished properties, the application of SNNs in the computationally intensive field of image generation is still under exploration. In this paper, we propose the Spiking Diffusion Models (SDMs), an innovative family of SNN-based generative models that excel in producing high-quality samples with significantly reduced energy consumption. In particular, we propose a Temporal-wise Spiking Mechanism (TSM) that allows SNNs to capture more temporal features from a bio-plasticity perspective. In addition, we propose a threshold-guided strategy that can further improve the performances by up to 16.7% without any additional training. We also make the first attempt to use the ANN-SNN approach for SNN-based generation tasks. Extensive experimental results reveal that our approach not only exhibits comparable performance to its ANN counterpart with few spiking time steps, but also outperforms previous SNN-based generative models by a large margin. Moreover, we also demonstrate the high-quality generation ability of SDM on large-scale datasets, e.g., LSUN bedroom. This development marks a pivotal advancement in the capabilities of SNN-based generation, paving the way for future research avenues to realize low-energy and low-latency generative applications. Our code is available at https://github.com/AndyCao1125/SDM.

With the growing prominence of the Mixture of Experts (MoE) architecture in developing large-scale foundation models, we investigate the Hierarchical Mixture of Experts (HMoE), a specialized variant of MoE that excels in handling complex inputs and improving performance on targeted tasks. Our analysis highlights the advantages of using the Laplace gating function over the traditional Softmax gating within the HMoE frameworks. We theoretically demonstrate that applying the Laplace gating function at both levels of the HMoE model helps eliminate undesirable parameter interactions caused by the Softmax gating and, therefore, accelerates the expert convergence as well as enhances the expert specialization. Empirical validation across diverse scenarios supports these theoretical claims. This includes large-scale multimodal tasks, image classification, and latent domain discovery and prediction tasks, where our modified HMoE models show great performance improvements compared to the conventional HMoE models.

Due to the recurrent structure of RNN, the long information propagation path poses limitations in capturing long-term dependencies, gradient explosion/vanishing issues, and inefficient sequential execution. Based on this, we propose a novel paradigm called Parallel Gated Network (PGN) as the new successor to RNN. PGN directly captures information from previous time steps through the designed Historical Information Extraction (HIE) layer and leverages gated mechanisms to select and fuse it with the current time step information. This reduces the information propagation path to O(1), effectively addressing the limitations of RNN. To enhance PGN's performance in long-range time series forecasting tasks, we propose a novel temporal modeling framework called Temporal PGN (TPGN). TPGN incorporates two branches to comprehensively capture the semantic information of time series. One branch utilizes PGN to capture long-term periodic patterns while preserving their local characteristics. The other branch employs patches to capture short-term information and aggregate the global representation of the series. TPGN achieves a theoretical complexity of O(L), ensuring efficiency in its operations. Experimental results on five benchmark datasets demonstrate the state-of-the-art (SOTA) performance and high efficiency of TPGN, further confirming the effectiveness of PGN as the new successor to RNN in long-range time series forecasting. The code is available in this repository: https://github.com/Water2sea/TPGN.

Mixture-of-experts (MoE) is becoming popular due to its success in improving the model quality, especially in Transformers. By routing tokens with a sparse gate to a few experts (i.e., a small pieces of the full model), MoE can easily increase the model parameters to a very large scale while keeping the computation cost in a constant level. Most existing works just initialize some random experts, set a fixed gating strategy (e.g., Top-k), and train the model from scratch in an ad-hoc way. We identify that these MoE models are suffering from the immature experts and unstable sparse gate, which are harmful to the convergence performance. In this paper, we propose an efficient end-to-end MoE training framework called EvoMoE. EvoMoE starts from training one single expert and gradually evolves into a large and sparse MoE structure. EvoMoE mainly contains two phases: the expert-diversify phase to train the base expert for a while and spawn multiple diverse experts from it, and the gate-sparsify phase to learn an adaptive sparse gate and activate a dynamic number of experts. EvoMoE naturally decouples the joint learning of both the experts and the sparse gate and focuses on learning the basic knowledge with a single expert at the early training stage. Then it diversifies the experts and continues to train the MoE with a novel Dense-to-Sparse gate (DTS-Gate). Specifically, instead of using a permanent sparse gate, DTS-Gate begins as a dense gate that routes tokens to all experts, then gradually and adaptively becomes sparser while routes to fewer experts. Evaluations are conducted on three popular models and tasks, including RoBERTa for masked language modeling task, GPT for language modeling task and Transformer for machine translation task. The results show that EvoMoE outperforms existing baselines, including Switch, BASE Layer, Hash Layer and StableMoE.