The paper investigates how momentum methods affect large‑batch training in a one‑pass setting using power‑law kernel regression. It derives critical learning rates for SGD, Polyak, and Nesterov, and shows how these rates depend on batch size, momentum, and model capacity. The authors provide scaling laws for risk dynamics, a three‑regime batch‑size phase diagram, and demonstrate that Polyak increases the critical batch size while Nesterov improves data efficiency in the large‑batch regime.
arXiv:2602. 11557v2 Announce Type: replace Abstract: A variety of widely used optimization methods like SignSGD and Muon can be interpreted as instances of steepest descent under different norm-induced geometries.
By Jichu Li, Xuan Tang, Difan Zou
The paper investigates how momentum methods affect large‑batch training in a one‑pass setting using power‑law kernel regression. It derives critical learning rates for SGD, Polyak, and Nesterov, and shows how these rates depend on batch size, momentum, and a capacity exponent. The authors then analyze risk dynamics, optimize final‑step risk under a fixed data budget, and present a three‑regime batch‑size phase diagram that highlights Polyak’s ability to enlarge the critical batch size and Nesterov’s superior data efficiency in the large‑batch regime.
By Jia-Nan Wang, Zixun Huang, Kairui Li, Lei Wu
arXiv:2605. 24316v3 Announce Type: replace Abstract: Mini-batching is central to large-scale optimization, yet its role in statistical scaling laws remains limited.
By Ziyan Chen, Zhongzhu Zhou, Ding-Xuan Zhou
arXiv:2510. 14717v2 Announce Type: replace-cross Abstract: Increasing the batch size during training -- a ''batch ramp'' -- is a promising strategy to accelerate large language model pretraining.
By Alexandru Meterez, Depen Morwani, Jingfeng Wu, Costin-Andrei Oncescu, Cengiz Pehlevan, Sham Kakade
arXiv:2602. 03001v2 Announce Type: replace-cross Abstract: To maximize hardware utilization, modern machine learning systems typically employ large constant or manually tuned batch size schedules, relying on heuristics that are brittle and costly to tune.
By Hiroki Naganuma, Shagun Gupta, Youssef Briki, Ioannis Mitliagkas, Irina Rish, Parameswaran Raman, Hao-Jun Michael Shi
arXiv:2607. 08104v1 Announce Type: new Abstract: Stochastic gradient descent (SGD) is a cornerstone of modern optimization.
By Ryusei Yamada, Naoki Sato, Hideaki Iiduka
arXiv:2508. 03105v3 Announce Type: replace Abstract: We analyze the convergence behavior of stochastic gradient descent with momentum (SGDM) under dynamic learning-rate and batch-size schedules by introducing a novel and simpler Lyapunov function.
By Yuichi Kondo, Hideaki Iiduka
Musec introduces MomentUm SpEctral Clipping, an optimizer-level, architecture‑agnostic technique that replaces Muon’s spectral flattening with selective spectral clipping to stabilize training. By clipping singular values above a threshold while preserving the momentum’s spectral structure, Musec addresses loss spikes and unbounded weight growth without requiring architecture‑specific changes. Soft Musec, an efficient implementation using smooth spectral saturation via coupled Newton‑Schulz iterations, offers convergence guarantees in nonconvex nonsmooth stochastic optimization and empirically improves stability across diverse learning rates and model sizes.
By Zhuanghua Liu, Menglian Wang, Luo Luo
arXiv:2601. 12238v5 Announce Type: replace-cross Abstract: In this paper, we provide a comprehensive theoretical analysis of Stochastic Gradient Descent (SGD) and its momentum variants (Polyak Heavy-Ball and Nesterov) for tracking time-varying optima under strong convexity and smoothness.
By Sharan Sahu, Cameron J. Hogan, Martin T. Wells
arXiv:2609.40148v1 Announce Type: new
Abstract: Power-law learning curves are often treated as fixed properties of a model and its data, although learning-rate and batch-size schedules can change the...
By Yichen Wang, Fanghui Liu, Yudong Chen
The paper proves that stochastic gradient descent with gradient clipping and additive Gaussian noise (SGD‑CN) converges almost surely under smoothness and bounded noise assumptions, given standard decaying step sizes. The analysis extends to momentum variants such as the stochastic heavy ball and Nesterov's accelerated gradient, showing that careful energy constructions yield similar guarantees. These results provide stronger theoretical foundations for understanding the pathwise behaviour of clipped stochastic gradient methods in both convex and nonconvex regimes.
By Amartya Mukherjee, Jun Liu