Convergence Analysis of SGD under Expected Smoothness

• URL: http://arxiv.org/abs/2510.20608v2
• Date: Sun, 26 Oct 2025 02:53:43 GMT
• Title: Convergence Analysis of SGD under Expected Smoothness
• Authors: Yuta Kawamoto, Hideaki Iiduka,
• Abstract summary: gradient descent (SGD) is the workhorse of large-scale learning, yet classical analyses rely on assumptions that can be either too strong (bounded variance) or too coarse (uniform noise)<n>The expected smoothness (ES) condition has emerged as a flexible alternative that ties the second moment of gradients to the objective value and the full gradient.
• Score: 7.2620484413601325
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Stochastic gradient descent (SGD) is the workhorse of large-scale learning, yet classical analyses rely on assumptions that can be either too strong (bounded variance) or too coarse (uniform noise). The expected smoothness (ES) condition has emerged as a flexible alternative that ties the second moment of stochastic gradients to the objective value and the full gradient. This paper presents a self-contained convergence analysis of SGD under ES. We (i) refine ES with interpretations and sampling-dependent constants; (ii) derive bounds of the expectation of squared full gradient norm; and (iii) prove $O(1/K)$ rates with explicit residual errors for various step-size schedules. All proofs are given in full detail in the appendix. Our treatment unifies and extends recent threads (Khaled and Richt\'arik, 2020; Umeda and Iiduka, 2025).

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