The Implicit Bias of Steepest Descent with Mini-batch Stochastic Gradient

• URL: http://arxiv.org/abs/2602.11557v1
• Date: Thu, 12 Feb 2026 04:25:38 GMT
• Title: The Implicit Bias of Steepest Descent with Mini-batch Stochastic Gradient
• Authors: Jichu Li, Xuan Tang, Difan Zou,
• Abstract summary: We study how batch size, momentum, and variance reduction shape the limiting max-margin behavior and convergence rates.<n>We show that without momentum, convergence only occurs with large batches, yielding a batch-dependent margin gap but the full-batch convergence rate.
• Score: 32.97211471008323
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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. In this work, we study the implicit bias of mini-batch stochastic steepest descent in multi-class classification, characterizing how batch size, momentum, and variance reduction shape the limiting max-margin behavior and convergence rates under general entry-wise and Schatten-$p$ norms. We show that without momentum, convergence only occurs with large batches, yielding a batch-dependent margin gap but the full-batch convergence rate. In contrast, momentum enables small-batch convergence through a batch-momentum trade-off, though it slows convergence. This approach provides fully explicit, dimension-free rates that improve upon prior results. Moreover, we prove that variance reduction can recover the exact full-batch implicit bias for any batch size, albeit at a slower convergence rate. Finally, we further investigate the batch-size-one steepest descent without momentum, and reveal its convergence to a fundamentally different bias via a concrete data example, which reveals a key limitation of purely stochastic updates. Overall, our unified analysis clarifies when stochastic optimization aligns with full-batch behavior, and paves the way for perform deeper explorations of the training behavior of stochastic gradient steepest descent algorithms.

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