← 研踪 PaperTrailarXiv:2510.20608 · v3
Convergence Analysis of SGD under Expected Smoothness
首次提交 2025-10-23 22:39
版本更新 2026-11-08 20:23
雷达观测 2026-11-10 12:16
摘要
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 sublinear convergence rates with explicit residual errors for various step-size schedules. Furthermore, we empirically validate our theoretical framework by training a ResNet-18 model on the CIFAR-100 dataset. We uncover a two-phase dynamic in the gradient noise structure, transitioning from a suboptimality-dominated regime to a geometry-dominated regime. Our treatment unifies and extends recent theoretical threads in the literature.
作者备注：50 pages, 11 figures, Accepted for publication in Journal of Optimization Theory and Applications (JOTA)
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v3 · 2026-11-08 20:23 · 最新观测版本
Convergence Analysis of SGD under Expected Smoothness
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 sublinear convergence rates with explicit residual errors for various step-size schedules. Furthermore, we empirically validate our theoretical framework by training a ResNet-18 model on the CIFAR-100 dataset. We uncover a two-phase dynamic in the gradient noise structure, transitioning from a suboptimality-dominated regime to a geometry-dominated regime. Our treatment unifies and extends recent theoretical threads in the literature.
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