Momentum Stochastic Gradient Descent

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Momentum

3 papers in the last four weeks, level with the four weeks before. 0.0% of all new papers.

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Latest papers 38

Oct 1, 2026math.OC

Convergence Analysis of STORM Under Different Geometries

Stochastic recursive momentum (STORM) achieves fast convergence for nonconvex optimization via the variance reduction effect, but existing analyses rely on the strong average smoothness assumption. In this paper, we study the convergence of STORM for different objectives without average smoothness. We first revisit the results under average smoothness, obtaining the

O(T^{-1/3})

bound for nonconvex objectives and the

O(σ^2/(μT))

bound for last-iterate output under the

μ

-Polyak--Łojasiewicz~(PL) condition. Without average smoothness, we design an auxiliary sequence and compare the STORM update with it in the analysis. With the help of this sequence, we prove that STORM still attains an

O(T^{-1/4})

rate for nonconvex objectives, which is optimal under standard smoothness. For convex and

λ

-strongly convex objectives, we further prove averaged and last-iterate bounds with optimal rates of

O(σR/\sqrt T)

and

O(σ^2/(λT))

, respectively. All the obtained results use the same STORM recursion with different hyperparameter choices.

Wei Jiang, Yibo Wang, Wenhao Yang +3

School of Computer Science and Engineering, Nanjing University of Science and Technology, Nanjing, China · School of Artificial Intelligence, Nanjing University, Nanjing, China

Stochastic Convex Optimization Convex Optimization Smoothness Convergence Momentum Stochastic Gradient Descent

Sep 30, 2026cs.LG

Convergence of Practical Muon with Finite Newton-Schulz Iterations and Nesterov Momentum

Practical Muon maintains momentum and performs a small, fixed number of Newton--Schulz iterations separately for each parameter matrix, often with a Nesterov correction. We analyze these layer-wise finite-step updates jointly on a coupled nonconvex objective, rather than replacing them by exact polar factors or one global orthogonalization. Under gradient-dependent

(\mathcal L_0,\mathcal L_1,q)

-smoothness and conditionally unbiased stochastic gradients with bounded layer-wise variance, we establish an

\mathcal O(T^{-1/4})

bound on the expected average Frobenius gradient norm. The analysis retains the Nesterov recursion and requires neither bounded stochastic gradients, symmetric noise, nor a uniform positive lower bound on the nonzero output singular values. Its constants contain no explicit matrix-dimension or rank factors when the number of blocks and problem constants are fixed. The proof follows a descent inequality and a decomposition of the momentum tracking error into initialization, noise, and drift. For the original five-step quintic, we verify the required scalar-map bounds analytically; the result also allows step-dependent coefficients satisfying the same bounds. A complementary nuclear-norm result quantifies rank dependence under a stronger spectral condition. The vanishing rate uses coupled learning-rate and momentum schedules, including the standard single-coefficient Nesterov rule.

Hanyng Peng, Hui Wang, Yue Yu

Pengcheng Laboratory, Shenzhen, China

Momentum Stochastic Gradient Descent Newton Muon Spectral Norm Convergence Iterative

Sep 28, 2026cs.LG

The Hidden Ratio in Adam: Stable Structure, Compression, and Sign Dynamics

Adam is the default optimizer for training modern deep neural networks, yet its adaptive behavior remains poorly understood due to the complex interaction between its first- and second-moment exponential moving averages (EMAs). We study Adam in the tied-

β

regime, where the two EMA decay rates are equal, and show that its adaptive dynamics can be expressed through a transformed ratio with approximately scale-stable behavior. Empirically, this transformed ratio exhibits a stable, heavy-tailed distribution across tasks, model scales, and training stages, in contrast to the variability of raw moment magnitudes. This empirical stability has both practical and conceptual consequences. First, we derive a recurrence for the transformed ratio, yielding a reparameterization of Adam that replaces the second moment with a compressible state. Leveraging its stable distribution, we show that a fixed 4-bit codebook is sufficient in our experiments to store this state without auxiliary scaling, achieving performance competitive with full-precision Adam. Second, the transformed ratio view clarifies Adam's connection to sign-based methods: Adam reduces to sign-based momentum modulated by the transformed ratio, and replacing it with a constant recovers Signum as a limiting case. This perspective further provides a simple rule for transferring learning rates between the two methods. Together, these results suggest that tied-

β

Adam admits a simple and approximately stable ratio structure underlying its adaptive behavior and demonstrate its utility for both analysis and efficient implementation.

Yihe Zhou, Tongtian Zhu, Yingxiao Huo +4

Department of Computer Science, The University of Manchester, Manchester, UK · College of Computer Science and Technology, Zhejiang University, Hangzhou, China

Adam Momentum Stochastic Gradient Descent Ratio Data Compression Methods Sign

Sep 27, 2026cs.RO

MomWorld: Momentum-Aware Latent World Model for Long-Horizon Autonomous Driving

Long-horizon planning enables autonomous vehicles to anticipate scene evolution and potential risks, supporting safe and stable decisions in complex interactions. However, existing methods struggle to propagate motion trends from observed history into the future. Long rollouts based on a single latent state may further attenuate useful dynamics, retain stale motion patterns, and disrupt reliable near-term plans. We introduce MomWorld, a momentum-aware latent world model for long-horizon planning. MomWorld extracts scene motion trends from historical-to-current observations and propagates latent momentum into future horizons, jointly predicting future configuration and momentum states. A learnable momentum persistence mechanism preserves stable trends, scene-conditioned momentum updates adapt future dynamics, and a scene-adaptive reset gate suppresses stale momentum under abrupt changes. We further propose MoFlow, a momentum-conditioned flow-matching module that refines a base trajectory to align with the predicted future scene evolution in only a few integration steps, with a horizon-aware residual fusion that preserves near-term planning stability while permitting stronger long-range corrections. Extensive experiments on NAVSIM, nuScenes and Bench2Drive demonstrate that MomWorld improves long-horizon planning consistency and reduces the average collision rate by 12.2% relative to MomAD over a 6-second planning horizon.

Ziying Song, Shengkai Zhang, Lei Yang +6

Nanyang Technological University · Beijing Jiaotong University · North University of China +2

Long-Horizon Planning Latent World Models Robot State World Models World Momentum Stochastic Gradient Descent

Sep 23, 2026cs.LG

ZO-COSMO: Index-Free One-Hop Mixing for Decentralized Zeroth-Order Optimization

Sparse communication in decentralized zeroth-order learning requires compatible peer-state coordinates. We characterize this one-hop condition and develop \textsf{ZO-COSMO}, coupling two-query estimation with average-preserving masked consensus using

q

values per active link. Global supports serve all-neighbor mixing; matching updates require agreement only within each pair. We derive a sharp contraction-per-scalar bound within the matching class and convergence guarantees for the core and sparse-momentum updates. At fixed matching, exact moment identities characterize how shared directions preserve gradient-heterogeneity cancellation and redistribute estimation error and disagreement. Mechanism experiments cover unequal curvatures, noise, and sparse momentum. Further tests span

64

synthetic agents and eight logical Qwen LoRA workers. At matched payload budgets, Qwen2-7B QNLI gains

3.65

accuracy points over explicit-index Rand-

k

; edge-local updates gain

3.42

and

2.53

points over all-neighbor mixing on eight-worker complete and ring graphs. A matched-first-step ablation gives a

3.92

-point momentum benefit. Seed-aware and same-matching controls distinguish encoding, scheduling, and query correlation.

Shengjun Zhang, Tingyi Liu, Heng Zhang +1

School of Artificial Intelligence, Hubei University, Wuhan, China · School of Economics and Management, Wuhan University, Wuhan, China · School of Electrical Engineering, Shanghai Jiao Tong University, Shanghai, China +1

Momentum Stochastic Gradient Descent

Sep 8, 2026cs.CV

ReMoMask-2: Latent Retrieval-Augmented Masked Motion Generation

Text-to-motion (T2M) generation maps natural language to human joint movements, aiding gaming, VR, and robotics. Retrieval-Augmented Text-to-Motion (RAG-T2M) improves generation on complex descriptions by conditioning on retrieved motion-text pairs. However, existing RAG-T2M models face two challenges: coarse-grained retrieval and fusion mechanisms overlook the hierarchical, spatial-temporal topology of human motion, and a representation gap exists because retrieved evidence resides in a semantic space separate from the generator's latents. To address the first, we present ReMoMask, a structure-aware RAG framework coupling Hierarchical Bidirectional Momentum (HBM) contrastive learning to align global and part-level features with text; Semantic Spatial-Temporal Attention (SSTA) for topology-aware fusion; and Topology Structured Masking (TSM) to force robust part-level grounding via adaptive masking. To address the second, we introduce ReMoMask-2, which rebuilds the retrieval database directly within the generator's pre-quantization latent space and aligns text queries via a distilled lightweight projector, allowing the generator to directly consume the retrieved motion's semantic content. Extensive experiments on HumanML3D, KIT-ML, and SnapMoGen demonstrate our retriever achieves state-of-the-art accuracy, while ReMoMask-2 attains the lowest FID on KIT-ML and SnapMoGen; notably, its single mask-transformer stage surpasses ReMoMask's full two-stage pipeline and delivers the fastest inference.

Yiran Wang, Zeyu Zhang, Ling Shao +1

University of Sydney, Camperdown 2006, Australia · School of Computer Science, Peking University, Beijing 100871, China · UCAS-Terminus AI Lab, University of Chinese Academy of Sciences, Beijing 101408, China

Text-To-Motion Generation Neural Mask Estimation Humanml3D Motion Momentum Stochastic Gradient Descent

Aug 19, 2026cs.LG

Activation-Keyed Momentum: An Anisotropic Momentum Update via the Delta Rule

Most modern optimizers form their momentum as an exponential moving average (EMA) of past gradients, forgetting every direction at one fixed rate. However, the inputs a deep network sees during training can be highly anisotropic, with a few directions queried frequently while most are seen rarely. Preconditioning methods address this anisotropy by wrapping extra processing around this buffer and leave the momentum update itself unchanged. We propose Activation-Keyed Momentum (AK-Momentum), which builds direction-awareness into the momentum update rule. The gradient of a linear layer splits into an input activation that acts as a key and an output-side error that acts as a value. Keying on that activation, AK-Momentum updates the momentum buffer by the canonical delta rule, so each direction is forgotten at a rate set by how often it appears. We prove that it is a valid momentum, that it applies the input-side curvature correction without matrix inversion, and that it clears stale directions faster than EMA under both a fixed and a drifting optimum. It is a drop-in replacement for the momentum buffer of any optimizer, its coefficient transfers across widths under

μ

P, and its extra compute stays between

22.2\%

and

25.0\%

of a gated-MLP block's linear cost with no persistent memory. In FineWeb-Edu pretraining, AdamW with AK-Momentum (AK-AdamW) reaches AdamW's validation loss in up to

46.39 \pm 4.32\%

fewer steps at 67M and

22.12 \pm 0.80\%

at 370M over three seeds, and the gain persists at 1B on a Chinchilla-optimal budget. A Muon baseline tuned under the same protocol sits above AK-AdamW at both language-model scales, and the gain holds for SGD, ResNet-18, and ViT-Tiny on CIFAR-10. Training-time diagnostics confirm the predicted mechanism, better gradient tracking and healthier input directions.

Euijin Hong, Guannan Qu

Electrical and Computer Engineering, Carnegie Mellon University

Momentum Stochastic Gradient Descent Momentum Gradient Deep Network

Aug 8, 2026cs.LG

Second Order Drifting Models

Drifting models are a recent class of one-step generative models that evolve the model distribution during training using a predefined sample-based drift field. Although they avoid iterative inference, their kernel-based drift fields induce frequency-dependent training dynamics: In the linearized regime, each Fourier mode of the density residual decays at a rate determined by the kernel spectrum, leading to slow recovery of fine-scale structure. We propose Second-Order Drifting Models, which lift drifting dynamics into phase space by augmenting generated samples with artificial velocity variables. We show that the resulting density perturbations obey accelerated second-order dynamics in Fourier space, connecting drifting models to the celebrated Nesterov acceleration from optimization theory. This provides a principled mechanism for mitigating the spectral stiffness of first-order drifting while preserving one-step inference. We derive a practical semi-implicit training algorithm and evaluate it on synthetic distribution matching, sequential data generation, and robotic control. Across these settings, the second-order drifting model improves convergence behavior and achieves competitive or superior performance over first-order drifting baselines.

Drake Brown, Yuhao Huang, Shih-Hsin Wang +1

Department of Mathematics Scientific Computing and Imaging (SCI) Institute University of Utah

Generative Models Momentum Stochastic Gradient Descent Order

Aug 6, 2026math.OC

An Inertial Block Proximal Linearized Method with Adaptive Momentum for Nonconvex and Nonsmooth Optimization

In this paper, we consider a class of multiblock nonconvex nonsmooth optimization problems, which covers many applications such as the analysis of pre-earthquake anomalies and machine learning. To solve this class of problems, we propose the inertial block proximal linearized method with two-phase adaptive momentum (IBPL

^+

-TP). Compared to the current methods, our method possesses three main advantages: (1) it introduces a two-phase adaptive momentum strategy to effectively update the extrapolation parameters, (2) it allows using two different extrapolation points to accelerate the convergence, (3) it allows the extrapolation parameters of these two extrapolation points to be independent of and unconstrained by all other parameters. While maintaining the above advantages, we prove that our method ensures the monotonic convergence of the objective function of this class of problems, and we also prove that the sequence generated by our method globally converges to a critical point, as well as establish the convergence rate of our method. To demonstrate the effectiveness of our method, we apply it to solve two nonconvex and nonsmooth machine learning problems, namely sparse nonnegative matrix factorization with

\ell_0

-constraints and sparse nonnegative CP decomposition with

\ell_0

-constraints. The numerical experimental results on solving these problems show that our method outperforms several state-of-the-art methods.

Weifeng Yang

Yunnan Earthquake Administration, Kunming 650224, China

Proximal Operators Stochastic Convex Optimization Convex Optimization Nonconvex Momentum Stochastic Gradient Descent Inertial+1

Jul 31, 2026math.OC

Sign compression for Muon: SignMuon, MuonSign, and the Limits of Error Feedback

SignMuon compresses the Muon update to one bit per parameter by taking its elementwise sign, providing the most direct way to run a matrix-aware optimizer under an extremely low communication budget. It outperforms SignSGD in practice, yet it can ascend even on a linear function. Signing the gradient before the Linear Minimization Oracle (LMO), rather than after, does not repair this: we construct a small explicit instance on which sign-before (MuonUSign) and sign-on-both-sides (MuonSign) ascend as well, so no placement of the sign around the oracle descends in general. Error feedback, the standard remedy for a biased compressor, does not rescue SignMuon: when applied to Muon's output, error feedback can fail for every smoothness constant, step size, and momentum. Applied to the gradient, error feedback does work, and EF21-MuonUSign and EF21-MuonSign attain the standard

\mathcal{O}(T^{-1/2})

rate for the squared gradient norm on smooth nonconvex problems, the latter at one bit in each direction. Experiments then reverse the ordering: across centralized CIFAR-10, federated CIFAR-10, and the nanoGPT speedrun, the strongest compressed method is consistently sign-after-the-LMO, precisely the placement we prove divergent, with the provably convergent variants trailing it. Compressing after the LMO, a heuristic, matters more at these scales than the guarantee does.

Maria Smirnova, Alexey Kravatskiy

MIRIAI Moscow, Russia

Muon Momentum Stochastic Gradient Descent Data Compression Methods Feedback

Jul 25, 2026math.OC

Nesterov acceleration in optimizing over probability measures

Optimization over probability measures has become an increasingly important paradigm in modern machine learning, scientific computing, and uncertainty quantification. Motivated by Nesterov's accelerated gradient method in Euclidean space, we develop Heavy-ball and Nesterov acceleration methods over the probability measure space

\mathcal{P}_2

and establish non-asymptotic convergence guarantees that match their Euclidean counterparts. In particular, we derive convergence rates with respect to both the number of iterations and the number of particles used to represent the underlying probability distributions. Extending accelerated optimization from Euclidean space to probability measures is challenging. The natural notion of momentum requires concepts such as tangent bundles of the set of probability space and they are hard to operate numerically. To overcome these difficulties, we introduce two complementary lifting procedures. The first lifts probability measures to phase space through a Hamiltonian formulation, introducing momentum variables into the dynamics. The second lifts probability measures to a common Hilbert space, restoring the linear structure required for convergence analysis while simultaneously yielding executable particle dynamics. Together, these two complementary lifting procedures provide a systematic methodology for designing, analyzing, and implementing momentum-based accelerated optimization methods over probability measure spaces.

Jiaqi Tang, Qin Li, Wilfrid Gangbo

Department of Statistics, University of Wisconsin–Madison, Madison, WI 53706, USA · Department of Mathematics, University of Wisconsin–Madison, Madison, WI 53706, USA · Department of Mathematics, University of California, Los Angeles, Los Angeles, CA 90095, USA

Momentum Stochastic Gradient Descent Probability Measures Kernel Hilbert Spaces Particle Dynamics Measures

Jun 17, 2026cs.LG

Compute Efficiency and Serial Runtime Tradeoffs for Stochastic Momentum Methods

Stochastic momentum methods such as heavy ball (HB), Nesterov momentum, and variants of Accelerated SGD (ASGD) [Kidambi et al., 2018] are widely used in modern training, but their stochastic benefits depend on two distinct quantities: serial runtime, the number of iterations needed to reach a target accuracy, and compute efficiency (CE), the inverse total gradient-query or FLOP cost. Larger batches reduce serial runtime without hurting CE only when the contraction gap grows linearly with batch size. We study stochastic HB and ASGD for consistent linear regression with Gaussian covariates and prove finite-dimensional, discrete-time lower bounds on their batch-size tradeoffs. Our first result shows that HB does not improve the CE frontier over SGD for arbitrary spectra; rather, it preserves SGD-level CE over a larger batch-size window, allowing larger batches to reduce serial runtime until HB reaches its deterministic accelerated scale. This window can be a factor

\sqrtκ

larger than the SGD critical batch size. For ASGD, the picture is more spectrum-dependent: for rapidly decaying power-law spectra, ASGD improves small-batch CE over HB/SGD, but as batch size grows it trades this CE advantage for improved serial runtime. Synthetic linear-regression experiments verify these qualitative regimes, including near-overlap of ASGD and HB for slowly decaying spectra and the predicted CE--serial tradeoff for rapidly decaying spectra.

Depen Morwani, Alexandru Meterez, Pranav Nair +1

1Harvard University · 2Kempner Institute at Harvard University · †: Work done while at Harvard.

Stochastic Gradient Descent Momentum Stochastic Gradient Descent Batch Efficiency Trade-Off

Jun 15, 2026cs.LG

Scaling Adaptive Depth with Norm-Agnostic Residual Networks

Residual architectures are ubiquitous in deep learning, but they suffer from a subtle structural limitation: the norm of the residual stream can grow rapidly with depth. As a result, updates from later layers become small relative to the accumulated residual state. This reduces their impact on the representation and limits the benefits of scaling models in depth. To address this, we introduce NAG, a norm-agnostic residual architecture that separates magnitude from directional information in the residual stream, preserving meaningful layer contributions throughout depth and preventing later updates from being systematically suppressed by residual-norm growth. Importantly, NAG introduces only a negligible number of additional parameters and relies on simple operations that are easily kernel-fusible, preserving training efficiency in practice. We show that this architecture outperforms baseline Transformers, with gains that increase substantially as depth grows, enabling effective training of much deeper models. The norm-agnostic formulation also leads to an interpretable Mixture-of-Depths (MoD) mechanism that adaptively skips both attention and MLP layers. Beyond serving as a post-training accuracy-compute tradeoff, this mechanism can be used as a pretraining-time scaling strategy: under iso-FLOP training, compute saved by reducing per-token forward-pass cost can be reinvested into training on more tokens while keeping the total parameter count and KV-cache budget fixed. In our experiments, moderate Mixture-of-Depths rates of approximately 20%-25% match full-depth baseline performance under equal training compute while substantially reducing the number of executed layer parameters and forward-pass FLOPs. These results identify sparsity in depth as a new scaling axis for fixed-compute training, enabling very deep yet FLOP-efficient models.

Tomás Figliolia, Beren Millidge

Zyphra San Francisco, CA

Residual Networks Scaling Momentum Stochastic Gradient Descent

Jun 11, 2026cs.AI

A Three-Layer Framework for AI in Scientific Discovery

Current discussions of AI in scientific discovery are often dominated by two visible capabilities: search over existing knowledge and execution through optimization, simulation, and automation. Both are important, but neither fully captures the central act of discovery: the formation and evolution of models. This paper proposes a three-layer view of AI in discovery. Layer 1 is search and retrieval by large language models. Layer 2, as the main innovation of this paper, is model formation through qualitative reasoning: the capacity to recognize when a current framework is structurally inadequate and to understand the problem within a broader representational space, not through trial and error, but through structural insight into what is missing and where it can be found. Layer 3 is execution, optimization, and refinement. The main claim is that Layer 2 is both the most important and the least developed. Search without model formation remains confined to inherited frameworks, while execution without conceptual revision only amplifies an existing formulation. We illustrate Layer 2 reasoning through three case studies: S. S. Chern's intrinsic proof of the Gauss-Bonnet theorem, the resolution of the Nesterov Accelerated Gradient convergence problem via Lyapunov functions, and the autonomous disproof of the Erdos unit distance conjecture by OpenAI in 2026. Each case exhibits the same structural signature: a framework that had become inadequate, a missing conceptual object, and a resolution found in an unexpected neighboring field.

Guojun Liao

Department of Mathematics, University of Texas at Arlington

Scientific Discovery Discovery Momentum Stochastic Gradient Descent

Jun 7, 2026math.OC

OptMuon: Closed-Loop Orthogonalized Momentum Methods for Stochastic Optimization with Zero-Noise Optimality

Orthogonalized momentum updates, as used in Muon-style optimizers, have recently shown strong empirical stability in large-scale deep learning. However, most current orthogonalized methods are still paired with fixed, externally scheduled, or otherwise open-loop magnitude rules, so their scale is not directly calibrated from the realized optimization trajectory. Motivated by the closed-loop perspective behind Lipschitz-free and noise-adaptive methods, we propose OptMuon, a family of adaptive momentum orthogonalization methods for stochastic nonconvex optimization. OptMuon combines Muon-style polar-factor directions with a trajectory-dependent AdaGrad-Norm-type coefficient schedule, so that the update magnitude is determined by the observed gradient and momentum history rather than by a prescribed Lipschitz-dependent rule. The schedule does not use the smoothness constant, the variance level, or the bounded-gradient constant in parameter selection, and its running-maximum correction prevents isolated gradient spikes from causing excessive coefficient collapse. Under lower-boundedness, unbiased stochastic gradients with bounded variance, smoothness, and an almost-sure bounded stochastic-gradient condition, we prove two complementary expected-stationarity guarantees. OptMuon-A achieves the noise-adaptive rate

\tilde{\mathcal O}(T^{-1/2}+σ^{1/2}T^{-1/4})

under average smoothness, while OptMuon-I achieves

\tilde{\mathcal O}(T^{-1/2}+σ^{1/3}T^{-1/3})

under individual smoothness. In the zero-noise regime, both bounds automatically reduce to a nearly optimal deterministic first-order rate

\tilde{\mathcal O}(T^{-1/2})

without manual hyperparameter retuning. These results show that closed-loop scalar adaptation can be combined with Muon-style momentum orthogonalization while retaining noise adaptivity and zero-noise optimality up to logarithmic factors.

Ganzhao Yuan

Faculty of Computer Science and Artificial Intelligence Shenzhen University of Advanced Technology (SUAT), China

Muon Momentum Stochastic Gradient Descent Stochastic Convex Optimization Stochastic Optimization Closed Loop

Jun 3, 2026stat.ML

ReSGA: A Large Tail Risk Model for Learning Value-at-Risk and Expected Shortfall

Learning Value-at-Risk (VaR) and Expected Shortfall (ES) is important for managing financial risks effectively. Existing approaches with limited parameters are vulnerable to model misspecification in the era of big data. To address this limitation, we propose a large tail risk model, the retrieval-enhanced self-grouping autoencoder (ReSGA), which is designed with millions of parameters to exploit the rich cross-sectional dependence and long-term temporal dynamics of assets using their characteristics. Applied to monthly US equity returns from 1926 to 2023 with 153 firm characteristics, ReSGA outperforms twelve econometric and machine learning competitors in terms of out-of-sample loss and statistical backtesting. In addition, its forecast advantages can translate into significant economic gains from long-short decile portfolios that are constructed by a new size-enhanced left-side momentum strategy. To clarify the role of complexity, we further conduct a systematic scaling analysis and demonstrate that improvements in joint VaR-ES forecasting are primarily driven by data complexity rather than model complexity. Finally, our analyses of group-importance and transfer-learning exhibit the interpretability and cross-market generalizability of ReSGA.

Yichi Zhang, Ke Zhu, Zhoufan Zhu

Department of Statistics and Actuarial Science, The University of Hong Kong, Hong Kong · Wang Yanan Institute for Studies in Economics, Xiamen University, China · Department of Finance, School of Economics, Xiamen University, China

Conditional-Value-At-Risk Long-Tailed Distribution Risk Momentum Stochastic Gradient Descent

May 27, 2026stat.ML

Dynamics of Stochastic Momentum with Sparse Updates in High Dimensions

Existing theory of momentum assumes that gradients arrive at every parameter at a roughly constant rate, an assumption violated in practice by heavy-tailed data distributions and modern architectures. We theoretically analyze the dynamics of two tractable models of momentum under sparse updates: a least squares model with sparse inputs and a logistic regression model with a rare class. Both admit exact closed-form second-moment dynamics whose high-dimensional limits we characterize across three scaling exponents for sparsity, batch size, and momentum decay. The phase structure on both problems is governed by the ratio of two intrinsic timescales: a momentum retention timescale (how many active updates the buffer survives) and a learning timescale (how many active updates it takes to reduce the squared error). When learning is much slower than retention, the limit matches SGD; when learning is faster, the system is unstable; where the timescales coincide, we recover classical heavy-ball dynamics. The oscillatory dynamics occur at different momentum values for different token sparsity, creating a spectral conflict for global momentum across token frequencies.

Katie Everett, Elliot Paquette

Google DeepMind & MIT · McGill University & Mila

Momentum Stochastic Gradient Descent Momentum Learning Dynamics Oscillatory Dynamics Hamiltonian

May 26, 2026cs.LG

MONA: Muon Optimizer with Nesterov Acceleration for Scalable Language Model Training

The Muon optimizer has recently offered a promising alternative to AdamW for large language model training, leveraging matrix orthogonalization to produce geometry-aware updates. However, like all first-order methods, Muon can become trapped in sharp local minima. In this work, we present MONA, an optimizer that bridges Muon's orthogonalization framework with curvature-aware acceleration. MONA adds an acceleration term directly into Muon's gradient processing pipeline. This term is calculated from the exponential moving average of gradient differences. We provide a detailed convergence analysis for MONA, showing that the acceleration term introduces curvature-sensitive corrections while preserving Muon's spectral-norm regularization. Empirically, MONA achieves better convergence and downstream task performance compared to both Muon and AdamW across three scales of Mixture-of-Experts pretraining, spanning from 1B to 68B parameters, with the largest model trained on 1 trillion tokens. Furthermore, we conduct supervised fine-tuning on the MOE-68B-A3B model and evaluate it on general capability, mathematical reasoning, and code generation benchmarks, where MONA achieves SOTA performance.

Jiacheng Li, Jianchao Tan, Hongtao Xu +5

Meituan, Beijing, China

Muon Momentum Stochastic Gradient Descent Large Language Model Training Curvature-Aware Spectral Framework Orthogonality

May 25, 2026cs.LG

EMA-Nesterov: Stabilizing Nesterov's Lookahead for Accelerated Deep Learning Optimization

Lookahead-based acceleration methods, such as Nesterov's momentum, are widely used in optimization, but they often become unreliable in deep learning training mainly due to stochastic gradient noise and non-convex loss landscapes. In particular, standard lookahead relies on short-horizon update signals (e.g., differences between consecutive iterates), which are inherently noisy and can lead to unstable extrapolation directions. This work revisits Nesterov's acceleration from a trajectory perspective and argues that effective acceleration in deep learning should harness the low-frequency trends of optimization trajectories rather than extrapolating noisy one-step updates. Leveraging this insight, we propose EMA-Nesterov, a simple modification that replaces the standard Nesterov's lookahead direction with an exponential moving average (EMA) of parameter updates. This yields a stabilized lookahead direction that captures and harnesses the evolving trend of the training trajectory through a low-pass filter, while remaining adaptive to progressive changes via the geometric weighting structure of EMA. We show that EMA-Nesterov retains a theoretical accelerated convergence rate in convex problems that is analogous to Nesterov's accelerated gradient method. Furthermore, we provide empirical evidence on language model pre-training to verify that EMA-Nesterov is broadly applicable across a range of fine-tuned base optimizers, including Adam, SOAP, Muon, as well as complex optimizers that achieve state-of-the-art performance on optimization benchmarks (NanoGPT). Compared to prior lookahead methods, EMA-Nesterov achieves better performance by avoiding the instability of short-horizon lookahead and the non-adaptivity of long-horizon lookahead.

Chung-Yiu Yau, Dawei Li, Athanasios Glentis +3

University of Minnesota · 2Amazon AGI · 3The Chinese University of Hong Kong

Momentum Stochastic Gradient Descent Neural Network Optimization Lookahead Deep Learning

May 19, 2026cs.LG

Ada2MS: A Hybrid Optimization Algorithm Based on Exponential Mixing of Elementwise and Global Second-Moment Estimates

Optimization algorithms are core methods by which machine learning models iteratively minimize loss functions, update parameters, learn from data, and improve performance. Momentum SGD and AdamW represent two important optimization paradigms. AdamW produces stable updates and usually has strong robustness across training scenarios, but its generalization performance is sometimes weaker than that of momentum methods. Momentum SGD can often obtain better generalization after careful tuning, but it is more sensitive to gradient-scale variation and hyperparameter settings. To balance the strengths and weaknesses of the two paradigms, this paper proposes Ada2MS, an optimization algorithm that achieves a smooth transition between AdamW-like behavior and momentum-SGD-like behavior through continuous exponential interpolation between elementwise second-moment estimates and global second-moment estimates. On the visual tasks evaluated in this study, Ada2MS obtains competitive results under a unified optimizer-comparison protocol. The code will be released at https://github.com/mengzhu0308/Ada2MS

Meng Zhu, Quan Xiao, Weidong Min

School of Information Management and Mathematics, Jiangxi University of Finance and Economics, Nanchang 330032, China · School of Mathematics and Computer Science, Nanchang University, Nanchang 330031, China · Institute of Metaverse, Nanchang University, Nanchang 330031, China +1

Adam Momentum Stochastic Gradient Descent Parameter-Efficient Adaptation

May 19, 2026cs.LG

MiMuon: Mixed Muon Optimizer with Improved Generalization for Large Models

Matrix-structured parameters frequently appear in many artificial intelligence models such as large language models. More recently, an efficient Muon optimizer is designed for matrix parameters of large-scale models, and shows markedly faster convergence than the vector-wise algorithms. Although some works have begun to study convergence properties (i.e., optimization error) of the Muon optimizer, its generalization properties (i.e., generalization error) is still not established. Thus, in this paper, we study generalization error of the Muon optimizer based on algorithmic stability and mathematical induction, and prove that the Muon has a generalization error of

O\big(\frac{1}{Nκ^{T}}\big)

, where

N

is training sample size, and

T

denotes iteration number, and

κ>0

denotes minimum difference between singular values of gradient estimate. To enhance generalization of the Muon, we propose an effective mixed Muon (MiMuon) optimizer by cautiously using orthogonalization of gradient, which is a hybrid of Muon and momentum-based SGD optimizers. Then we prove that our MiMuon optimizer has a lower generalization error of

O\big(\frac{1}{N}\big)

than

O\big(\frac{1}{Nκ^{T}}\big)

of Muon optimizer, since

κ

generally is very small. Meanwhile, we also studied the convergence properties of our MiMuon algorithm, and prove that our MiMuon algorithm has the same convergence rate of

O(\frac{1}{T^{1/4}})

as the Muon algorithm. Some numerical experimental results on training large models including Qwen3-0.6B and YOLO26m demonstrate efficiency of the MiMuon optimizer.

Feihu Huang, Yuning Luo, Songcan Chen

College of Computer Science and Technology, Nanjing University of Aeronautics and Astronautics, Nanjing, China · MIIT Key Laboratory of Pattern Analysis and Machine Intelligence, Nanjing, China · College of Design and Engineering, National University of Singapore, Singapore

Muon Momentum Stochastic Gradient Descent Improved Generalization

May 18, 2026cs.LG

Perfect Parallelization in Mini-Batch SGD with Classical Momentum Acceleration

Accelerating stochastic gradient methods with classical momentum schemes, such as Polyak's heavy ball, has proven highly successful in training large-scale machine learning models, particularly when combined with the hardware acceleration of large mini-batch computations. Yet, the effect of classical momentum on stochastic mini-batch optimization has been poorly understood theoretically, with prior works requiring strong noise assumptions and extremely large mini-batches. In this work, we develop a general theory of stochastic momentum acceleration for optimizing over quadratics in the interpolation regime, a popular abstraction for studying deep learning dynamics which also includes classical methods such as randomized Kaczmarz and coordinate descent. Our framework encompasses both heavy ball and Nesterov-style momentum, allows for arbitrary mini-batch sizes, and makes minimal assumptions on the stochastic noise. In particular, we show that acceleration from classical momentum is directly proportional to the gradient mini-batch size (up to a natural saturation point), thereby enabling perfect parallelization of mini-batch computations. Our theory also provides a simple choice for the momentum parameter, which is shown to be effective empirically.

Sachin Garg, Michał Dereziński

University of Michigan

Momentum Stochastic Gradient Descent Stochastic Gradient Descent Batch Momentum Acceleration Learning Dynamics

May 18, 2026cs.LG

Ringmaster LMO: Asynchronous Linear Minimization Oracle Momentum Method

Muon has recently emerged as a strong alternative to AdamW for training neural networks, with encouraging large-scale pretraining results and growing evidence that matrix-structured updates can be faster in practice. Yet Muon, and more generally Linear Minimization Oracle (LMO) based methods, are typically used synchronously. This is problematic in heterogeneous distributed systems, where workers complete gradient computations at different speeds and synchronous training must repeatedly wait for slower workers. In this work, we introduce Ringmaster LMO, an asynchronous LMO-based momentum method for unconstrained stochastic nonconvex optimization. Our method builds on the delay-thresholding idea of Ringmaster ASGD. For SGD-type methods, Ringmaster ASGD achieves optimal time complexity by discarding overly stale gradients. Ringmaster LMO extends this mechanism to general LMO-based updates. We establish convergence guarantees under generalized

(L_0, L_1)

-smoothness and further develop a parameter-agnostic variant with decreasing stepsizes and adaptive delay thresholds. Finally, we translate our iteration guarantees into time complexity bounds under heterogeneous worker computation times. In the classical Euclidean smooth setting, these bounds recover the optimal time complexity of Ringmaster ASGD. Experiments on stochastic quadratic problems and NanoChat language-model pretraining show that the advantages of Ringmaster LMO grow with system heterogeneity and that the method outperforms strong synchronous and asynchronous baselines.

Abdurakhmon Sadiev, Artavazd Maranjyan, Ivan Ilin +1

KAUST

Momentum Stochastic Gradient Descent Muon Gradient

May 14, 2026math.OC

Stochastic Compositional Optimization via Hybrid Momentum Frank--Wolfe

Stochastic compositional optimization minimizes objectives of the form

\min_{\bm{x} \in \mathcal{X}} F(\bm{f}(\bm{x}), \bm{x})

, where

\bm{f}

is accessible only through noisy stochastic queries. Existing methods for this problem assume that the outer function

F

is continuously differentiable, which excludes many practically important applications such as robust max-of-losses, Conditional Value-at-Risk, and norm regularizers. We propose the Hybrid Momentum Stochastic Frank--Wolfe algorithm, which drops the smoothness assumption on

F

. By combining a momentum-based Jacobian tracker with a Taylor-corrected function tracker, the algorithm feeds an entire stochastic linearization -- rather than a single gradient -- into a generalized linear minimization oracle. We establish an

\mathcal{O}(K^{-1/4})

convergence rate in the generalized Frank--Wolfe gap for non-convex objectives with

L_F

-Lipschitz outer functions, matching the optimal complexity for projection-free single-sample stochastic methods under expected smoothness. The analysis extends to heavy-tailed noise oracles with bounded

r

-th moments for

r \in (1, 2]

and recovers the deterministic rates of Vladarean et al (2023) as the noise vanishes.

El Mahdi Chayti

Machine Learning and Optimization Laboratory (MLO) EPFL, Switzerland

Stochastic Convex Optimization Stochastic Optimization Momentum Stochastic Gradient Descent Stochastic Momentum

May 9, 2026cs.LG

Cosine-Gated Adam-Decay: Drop-In Staleness-Aware Outer Optimization for Decoupled DiLoCo

Asynchronous DiLoCo systems may receive pseudo-gradients computed several outer rounds earlier, yet the standard Nesterov outer optimizer does not explicitly condition its update on per-update age. This can make the outer momentum buffer brittle under large controlled delays. We propose Cosine Gated Adam Decay (CGAD), a simple, drop-in, age-aware outer optimizer that scales each incoming pseudo-gradient by

σ(τ) = γ(τ) e^{-ατ}

before it enters Adam's first- and second-moment buffers; the exponential models information decay and the cosine gate

γ(τ)

smoothly zeroes contributions past a chosen cutoff. CGAD reduces to plain Adam at

τ=0

, adds two hyperparameters whose defaults transfer across scales, and extends to partial-sync schedulers via a per-fragment age-aware variant (PA-CGAD). For an idealized gated-adaptive update on smooth non convex objectives, we prove a non-asymptotic convergence bound whose staleness-bias term depends on

α

alone, rather than on the realized maximum delay

τ_{\max}

; standard analyses of asynchronous momentum-SGD instead carry a

τ_{\max}^2

factor. Empirically, on Llama style language model pretraining at 25M, 1B, and 7B parameters, CGAD trains stably across the controlled delays we sweep. The cosine cutoff acts as scale insurance: the closest baseline, Adam Decay (CGAD without the cutoff), is competitive at 25M but its seed-to-seed

σ

at

τ=8

grows 27x from 25M to 7B, pushing its single-shot risk (mean +

σ

) above the chance-level loss while CGAD's stays well below. The published Nesterov recipe is the least stable method on the full sweep.

Vatsal Shah, Jiahao Sun

Adam Momentum Stochastic Gradient Descent Asynchronous Execution Decoupling Prompt Optimization

May 8, 2026math.OC

A Unified Lyapunov-IQC Framework for Uniform Stability of Smooth Quadratic First-Order Accelerated Optimizers

We develop a unified Lyapunov-integral quadratic constraint (IQC) framework for establishing uniform stability of first-order accelerated optimization algorithms in the

β

-smooth and

γ

-strongly convex regime. Classical analyses of uniform stability, such as the work of Hardt, Recht, and Singer for stochastic gradient descent (SGD), rely on direct coupling arguments and case-by-case control of iterate differences under random sampling. Extending such arguments to accelerated methods, such as Nesterov Accelerated Gradient (NAG), is complicated by the presence of higher-order state dynamics induced by momentum. We first extend this classical approach with the use of Lyapunov functions to provide a uniform stability bound for smooth quadratic NAG, and supplement this result with small-scale numerical experiments. We then extend this framework by modeling first-order accelerated optimizers as Lur'e-type feedback interconnections between a linear dynamical system and a (non-linear) gradient operator.

β

-Smoothness and

γ

-strong convexity are encoded a sector IQC inequality. Under this representation, uniform stability is certified via the existence of a quadratic Lyapunov function satisfying a finite-dimensional linear matrix inequality (LMI) in the form of a feasibility problem, which can be solved via semi-definite programming (SDP). We instantiate this framework for NAG and show how classical uniform stability bounds can be recovered via this framework. These results underscore a structural connection between optimization dynamics and robust control theory, providing a modular methodology for reliable and reproducible numerical certification of uniform stability and generalization behavior of first-order methods via convex optimization tools that is adaptable to increasingly complex optimization algorithms.

Don Li, Dacian Daescu

Department of Mathematics & Statistics, Portland State University, Portland, OR 97201, USA

Lyapunov Function Momentum Stochastic Gradient Descent Nonconvex

May 7, 2026math.OC

Muon with Nesterov Momentum: Heavy-Tailed Noise and (Randomized) Inexact Polar Decomposition

Most first-order optimizers treat matrix-valued parameters as vectors, ignoring the intrinsic geometry of hidden-layer weights in neural networks. Muon addresses this mismatch by updating along the polar factor of a momentum matrix, but its theoretical understanding has lagged behind practice. In particular, practical implementations incorporate Nesterov momentum, compute the polar factor only approximately, and operate with stochastic gradients that may be heavy-tailed. We close this gap by developing a convergence theory for Muon with Nesterov momentum and inexact polar decomposition in non-convex matrix optimization under heavy-tailed noise. Our analysis builds on a unified framework for inexact polar decomposition that captures practical iterative approximations such as Newton-Schulz and quantifies how their errors propagate through the optimization dynamics. Under this framework, we establish an optimal iteration and sample complexity of

O \left(\varepsilon^{\frac{-(3α-2)}{(α-1)}} \right)

for finding an

\varepsilon

-stationary point, where

α\in(1,2]

denotes the heavy-tail index. For the inexact-polar setting with

σ_1=0

, we also provide guarantees that do not require prior knowledge of

α

. We analyze a randomized low-rank polar decomposition that is substantially more efficient than full-space methods while remaining compatible with our theory. Numerical experiments further demonstrate the effectiveness of the proposed inexact and randomized variants.

Sayantan Choudhury, Xiaoran Cheng, Martin Takáč +2

MBZUAI · Penn State University · Georgia Institute of Technology +1

Momentum Stochastic Gradient Descent Muon Low-Rank Structure Long-Tailed Distribution

May 7, 2026cs.LG

MDN: Parallelizing Stepwise Momentum for Delta Linear Attention

Linear Attention (LA) offers a promising paradigm for scaling large language models (LLMs) to long sequences by avoiding the quadratic complexity of self-attention. Recent LA models such as Mamba2 and GDN interpret linear recurrences as closed-form online stochastic gradient descent (SGD), but naive SGD updates suffer from rapid information decay and suboptimal convergence in optimization. While momentum-based optimizers provide a natural remedy, they pose challenges in simultaneously achieving training efficiency and effectiveness. To address this, we develop a chunkwise parallel algorithm for LA with a stepwise momentum rule by geometrically reordering the update coefficients. Further, from a dynamical systems perspective, we analyze the momentum-based recurrence as a second-order system that introduces complex conjugate eigenvalues. This analysis guides the design of stable gating constraints. The resulting model, Momentum DeltaNet (MDN), leverages Triton kernels to achieve comparable training throughput with competitive linear models such as Mamba2 and KDA. Extensive experiments on the 400M and 1.3B parameter models demonstrate consistent performance improvements over strong baselines, including Transformers, Mamba2 and GDN, across diverse downstream evaluation benchmarks. Code: https://github.com/HuuYuLong/MomentumDeltaNet .

Yulong Huang, Xiang Liu, Hongxiang Huang +5

The Hong Kong University of Science and Technology (Guangzhou), Guangzhou, China

Kimi Delta Attention Linear Attention Large Language Model Training Momentum Stochastic Gradient Descent Gated Deltanet Momentum+1

May 4, 2026cs.LG

SignMuon: Communication-Efficient Distributed Muon Optimization

Distributed training of large neural networks is bottlenecked by full-precision gradient communication and by coordinatewise optimizers that ignore the matrix structure of weight tensors. We propose Sign-Muon, a 1-bit, matrix-aware optimizer that combines majority-vote sign aggregation from signSGD with the polar-step framework of Muon. Each worker forms a Muon-style direction by taking the polar factor of its momentum via a Newton--Schulz iteration, transmits only the entrywise signs, and aggregates by majority vote; an optional local polar step further enforces orthogonality at no extra communication cost. Under spectral-norm smoothness and bounded-variance stochastic gradients, the spectral-norm normalized sign step yields an

\mathcal{O}(1/\sqrt{T})

nonconvex rate for an

\ell_1

-based stationarity measure. With unimodal symmetric noise, majority vote across

M

workers cuts the stochastic term by

1/\sqrt{M}

, matching signSGD. In the

α

-

β

model, distributed Sign-Muon needs only one integer sum-allreduce per iteration; all orthogonalization is local, giving a

32\times

bandwidth reduction over float32 (

4\times

for int8). Across 330 CIFAR-10/ResNet-50 configurations Sign-Muon attains the best validation accuracy (92.15%); its 4-GPU majority-vote variant reaches 92.02% with 37% less training time at matched effective batch. On nanoGPT, Sign-Muon achieves lower perplexity and better anytime performance than other sign-based baselines, with favorable weak-scaling up to 16 GPUs.

Neel Mishra, Kushagara Trivedi, Pawan Kumar

1Microsoft · 2IIIT Hyderabad

Muon Decentralized Optimization Momentum Stochastic Gradient Descent Resnet-18 Majority Voting

May 1, 2026math.OC

Randomized Subspace Nesterov Accelerated Gradient

Randomized-subspace methods reduce the cost of first-order optimization by using only low-dimensional projected-gradient information, a feature that is attractive in forward-mode automatic differentiation and communication-limited settings. While Nesterov acceleration is well understood for full-gradient and coordinate-based methods, obtaining accelerated methods for general subspace sketches that use only projected-gradient information and can improve over full-dimensional Nesterov acceleration in oracle complexity is technically nontrivial. We develop randomized-subspace Nesterov accelerated gradient methods for smooth convex and smooth strongly convex optimization under matrix smoothness and generic sketch moment assumptions. The key technical ingredient is a three-sequence formulation tailored to matrix smoothness, which recovers the corresponding classical Nesterov methods in the full-dimensional case. The resulting theory establishes accelerated oracle-complexity guarantees and makes explicit how matrix smoothness and the sketch distribution enter the complexity. It also provides a unified basis for comparing sketch families and identifying when randomized-subspace acceleration improves over full-dimensional Nesterov acceleration in oracle complexity.

Gaku Omiya, Pierre-Louis Poirion, Akiko Takeda

Department of Mathematical Informatics, The University of Tokyo, Tokyo, Japan · Center for Advanced Intelligence Project, RIKEN, Tokyo, Japan

Momentum Stochastic Gradient Descent First Order Oracle Complexity Nonconvex

Apr 25, 2026stat.ML

Inference of Online Newton Methods with Nesterov's Accelerated Sketching

Reliable decision-making with streaming data requires principled uncertainty quantification of online methods. While first-order methods enable efficient iterate updates, their inference procedures still require updating proper (covariance) matrices, incurring

O(d^2)

time and memory complexity, and are sensitive to ill-conditioning and noise heterogeneity of the problem. This costly inference task offers an opportunity for more robust second-order methods, which are, however, bottlenecked by solving Newton systems with

O(d^3)

complexity. In this paper, we address this gap by studying an online Newton method with Hessian averaging, where the Newton direction at each step is approximately computed using a sketch-and-project solver with Nesterov's acceleration, matching

O(d^2)

complexity of first-order methods. For the proposed method, we quantify its uncertainty arising from both random data and randomized computation. Under standard smoothness and moment conditions, we establish global almost-sure convergence, prove asymptotic normality of the last iterate with a limiting covariance characterized by a Lyapunov equation, and develop a fully online covariance estimator with non-asymptotic convergence guarantees. We also connect the resulting uncertainty quantification to that of exact and sketched Newton methods without Nesterov's acceleration. Extensive experiments on regression models demonstrate the superiority of the proposed method for online inference.

Haoxuan Wang, Xinchen Du, Sen Na

School of Industrial and Systems Engineering, Georgia Institute of Technology.

Newton Momentum Stochastic Gradient Descent Uncertainty Quantification Fixed-Point Iteration Sketches

Jan 30, 2026cs.LG

Adaptive Momentum and Nonlinear Damping for Neural Network Training

Momentum Stochastic Gradient Descent (mSGD) relies on a fixed momentum coefficient shared across all parameters, failing to account for the heterogeneous structure of modern loss landscapes. In this work, we adopt a continuous-time formulation to introduce individual, adaptive momentum coefficients regulated by the kinetic energy of each model parameter. This mechanism automatically adjusts to evolving training dynamics to maintain stability without sacrificing convergence speed. We demonstrate that this adaptive friction is inextricably linked to cubic damping, a suppression mechanism from structural dynamics. We additionally introduce two optimization schemes by augmenting the continuous dynamics of mSGD and Adam with a cubic damping term. Empirically, our methods demonstrate robustness and match or outperform Adam on training ViT, BERT, and GPT2 tasks where mSGD typically struggles. We further provide theoretical results establishing the exponential convergence of the proposed schemes.

Aikaterini Karoni, Rajit Rajpal, Benedict Leimkuhler +1

School of Mathematics University of Bristol · School of Mathematics University of Edinburgh · CERMICS, CNRS, École des Ponts, Institut Polytechnique de Paris; Inria Paris, France

Momentum Stochastic Gradient Descent Neural Network Training Inverted Pendulum

Nov 4, 2025cs.LG

A Nesterov-Accelerated Byzantine-Robust Federated Learning

We investigate robust federated learning, where a group of workers collaboratively train a shared model under the orchestration of a central server in the presence of Byzantine adversaries capable of arbitrary and potentially malicious behaviors. To simultaneously enhance communication efficiency and resilience against such adversaries, we propose a Byzantine-resilient Nesterov-accelerated federated learning (Byrd-NAFL) algorithm. Byrd-NAFL seamlessly integrates Nesterov's momentum into the federated learning process alongside Byzantine-resilient aggregation rules to achieve fast and safe convergence against gradient corruption. We establish a finite-time convergence guarantee for Byrd-NAFL under non-convex and smooth loss functions with relaxed assumptions on the aggregated gradients. Extensive numerical experiments validate the effectiveness of Byrd-NAFL and demonstrate the superiority over existing benchmarks in terms of convergence speed, accuracy, and resilience to diverse malicious attacks.

Lihan Xu, Xiaoyi Fan, Gang Wang +3

School of Cyber Science and Engineering, Xi’an Jiaotong University, Xi’an 710049, China · Artificial Intelligence Research Institute and Guangdong-Hong Kong-Macao Joint Laboratory for Emotional Intelligence and Pervasive Computing, Shenzhen MSU-BIT University, Shenzhen 518172, China · State Key Laboratory of Autonomous Intelligent Unmanned Systems, Beijing Institute of Technology, Beijing 100081, China

Byzantine Attacks Federated Learning Momentum Stochastic Gradient Descent Convergence

Jul 16, 2025math.OC

Better Convergence Guarantees for Sign-Based Momentum Methods

This paper presents an improved analysis for sign-based methods with momentum updates. Traditional sign-based methods obtain a convergence rate of

\mathcal{O}(T^{-1/4})

under the separable smoothness assumption, but they typically require large batch sizes or assume unimodal symmetric stochastic noise. To address these limitations, we demonstrate that signSGD with momentum can achieve the same convergence rate using constant batch sizes without additional assumptions. We also establish a convergence rate under the

l_2

-smoothness condition, improving upon the result of prior work by a factor of

\mathcal{O}(d^{1/2})

, where

d

is the problem dimension. Furthermore, we explore sign-based methods in distributed settings and show that the proposed methods yield convergence rates of

\mathcal{O}\left( d^{1/2}T^{-1/2} + dn^{-1/2} \right)

and

\mathcal{O}\left(d^{1/4}T^{-1/4}\right)

, which outperform the previous results of

\mathcal{O}\left( dT^{-1/4} + dn^{-1/2} \right)

and

\mathcal{O}\left( d^{3/8}T^{-1/8} \right)

, respectively. Numerical experiments also validate the effectiveness of the proposed methods.

Wei Jiang, Dingzhi Yu, Sifan Yang +3

National Key Laboratory for Novel Software Technology, Nanjing University, Nanjing, China · School of Artificial Intelligence, Nanjing University, Nanjing, China

Momentum Stochastic Gradient Descent Convergence Smoothness Momentum Multiplicative Gamma Noise

Jul 4, 2025cs.LG

On the Effectiveness of the z-Transform Method in Quadratic Optimization

The z-transform of a sequence is a classical tool used within signal processing, control theory, computer science, and electrical engineering. It allows for studying sequences from their generating functions, with many operations that can be equivalently defined on the original sequence and its

z

-transform. In particular, the z-transform method focuses on asymptotic behaviors and allows the use of Taylor expansions. We present a sequence of results of increasing significance and difficulty for linear models and optimization algorithms, demonstrating the effectiveness and versatility of the z-transform method in deriving new asymptotic results. Starting from the simplest gradient descent iterations in an infinite-dimensional Hilbert space, we show how the spectral dimension characterizes the convergence behavior. We then extend the analysis to Nesterov acceleration, averaging techniques, and stochastic gradient descent.

Francis Bach

SIERRA · Inria - Ecole Normale Sup´erieure · PSL Research University

Momentum Stochastic Gradient Descent Convergence Stochastic Gradient Descent Fourier Kernel Hilbert Spaces

May 31, 2025cs.CL

Scaling Textual Gradients via Sampling-Based Momentum

LLM-based prompt optimization, which uses LLM-provided ``textual gradients'' (feedback) to refine prompts, has emerged as an effective method for automatic prompt engineering. However, its scalability and stability are unclear when using more data in training. We systematically investigate the potential and challenges of scaling training data in textual gradient descent. We show that naively scaling training examples is infeasible due to both explicit context-length limits and an implicit context wall, where long-context degradation yields diminishing returns. Inspired by prior wisdom in stochastic gradient descent, we propose Textual Stochastic Gradient Descent with Momentum (TSGD-M), which reweights updates through momentum sampling, using bootstrapped minibatch validation accuracy as importance weights over historical prompts. To stabilize TSGD and enable effective scaling within a limited context window, TSGD-M carries prior prompts information by \textit{dynamically} exploring the past top performing prompts without expanding input context length. TSGD-M integrates seamlessly into existing prompt optimization frameworks, including TextGrad, DSPy-COPRO, and AdalFlow, and achieves consistent gains across 6 benchmarks.

Zixin Ding, Junyuan Hong, Zhan Shi +6

University of Chicago · University of Texas at Austin · Santa Clara University +3

Large Language Model Training Automatic Prompt Optimization Stochastic Gradient Descent Text Generation Momentum Stochastic Gradient Descent Momentum+1

Oct 10, 2024cs.LG

How Learning Dynamics Drive Adversarially Robust Generalization?

Despite being widely adopted as a canonical framework for learning robust models, adversarial training suffers from robust overfitting. Existing empirical and theoretical explorations fail to provide a satisfactory mechanistic interpretation of the phenomenon. By modeling adversarial training with momentum SGD as a discrete-time dynamical system, we propose a PAC-Bayesian analytical framework that proves time-resolved robust generalization bounds. Specifically, our framework tracks the closed-form evolution of the posterior mean and covariance under both stationary and non-stationary transient regimes, connecting the model's robust generalization performance to learning rate, local loss geometry, and mini-batch stochastic gradients. By estimating the key quantities associated with the bound, we illustrate the underlying mechanism of robust overfitting. Our framework also shows how adversarial weight perturbation reduces robust generalization gaps by suppressing dominant loss-curvature modes, while suggesting that excessive penalization can be sub-optimal for optimization.

Yuelin Xu, Xiao Zhang

CISPA Helmholtz Center for Information Security, Saarbrücken, Germany

Adversarial Training Generalization Bounds Learnability Pac-Bayesian Theory Momentum Stochastic Gradient Descent

Jun 18, 2024cs.LG

Accelerated Stochastic Min-Max Optimization Based on Bias-corrected Momentum

Lower-bound analyses for nonconvex strongly-concave minimax optimization problems have shown that stochastic first-order algorithms require at least

\mathcal{O}(\varepsilon^{-4})

sample complexity to find an

\varepsilon

-stationary point. Some works indicate that this complexity can be improved to

\mathcal{O}(\varepsilon^{-3})

when the stochastic loss gradient is Lipschitz continuous. The question of achieving enhanced convergence rates under distinct conditions, remains open. In this work, we address this question for optimization problems that are nonconvex in the minimization variable and strongly concave or Polyak-Lojasiewicz (PL) in the maximization variable. We introduce novel bias-corrected momentum algorithms utilizing efficient Hessian-vector products. We establish convergence conditions and demonstrate a lower iteration complexity of

\mathcal{O}(\varepsilon^{-3})

for the proposed algorithms. The effectiveness of the proposed method is validated through applications to robust logistic regression and robust adaptive cruise control.

Haoyuan Cai, Sulaiman A. Alghunaim, Ali H. Sayed

Ecole Polytechnique Fédérale de Lausanne, Switzerland · Kuwait University, Kuwait

Stochastic Convex Optimization Convex Optimization Nonconvex Momentum Stochastic Gradient Descent Hessian Momentum+1