Stochastic Gradient Descent in Non-Convex Problems: Asymptotic Convergence with Relaxed Step-Size via Stopping Time Methods

• URL: http://arxiv.org/abs/2504.12601v1
• Date: Thu, 17 Apr 2025 02:56:20 GMT
• Title: Stochastic Gradient Descent in Non-Convex Problems: Asymptotic Convergence with Relaxed Step-Size via Stopping Time Methods
• Authors: Ruinan Jin, Difei Cheng, Hong Qiao, Xin Shi, Shaodong Liu, Bo Zhang,
• Abstract summary: Gradient Descent (SGD) is widely used in machine learning research.<n>This paper introduces a novel analytical method for convergence analysis of SGD under more relaxed step-size conditions.
• Score: 13.677904140815386
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Stochastic Gradient Descent (SGD) is widely used in machine learning research. Previous convergence analyses of SGD under the vanishing step-size setting typically require Robbins-Monro conditions. However, in practice, a wider variety of step-size schemes are frequently employed, yet existing convergence results remain limited and often rely on strong assumptions. This paper bridges this gap by introducing a novel analytical framework based on a stopping-time method, enabling asymptotic convergence analysis of SGD under more relaxed step-size conditions and weaker assumptions. In the non-convex setting, we prove the almost sure convergence of SGD iterates for step-sizes $ \{ \epsilon_t \}_{t \geq 1} $ satisfying $\sum_{t=1}^{+\infty} \epsilon_t = +\infty$ and $\sum_{t=1}^{+\infty} \epsilon_t^p < +\infty$ for some $p > 2$. Compared with previous studies, our analysis eliminates the global Lipschitz continuity assumption on the loss function and relaxes the boundedness requirements for higher-order moments of stochastic gradients. Building upon the almost sure convergence results, we further establish $L_2$ convergence. These significantly relaxed assumptions make our theoretical results more general, thereby enhancing their applicability in practical scenarios.

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