Step-Size Stability in Stochastic Optimization: A Theoretical Perspective

2026-02-10

AI Summary

This study investigates the sensitivity of stochastic optimization algorithms to stepsize choices and the resulting performance degradation. Through theoretical analysis, it establishes a quantitative relationship between stepsize and convergence bounds, providing the first direct theoretical evidence for the robustness of adaptive stepsize methods—such as SPS and NGN—relative to standard SGD, thereby moving beyond prior reliance on empirical comparisons alone. Both theoretical derivations and numerical experiments consistently demonstrate that adaptive methods exhibit significantly greater stability with respect to stepsize selection and more controllable performance degradation. This robustness holds in both convex and non-convex settings, with the latter underscoring the continued theoretical relevance of adaptive strategies even in challenging non-convex optimization landscapes.

Technology Category

• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Optimization

Application Category

• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

We present a theoretical analysis of stochastic optimization methods in terms of their sensitivity with respect to the step size. We identify a key quantity that, for each method, describes how the performance degrades as the step size becomes too large. For convex problems, we show that this quantity directly impacts the suboptimality bound of the method. Most importantly, our analysis provides direct theoretical evidence that adaptive step-size methods, such as SPS or NGN, are more robust than SGD. This allows us to quantify the advantage of these adaptive methods beyond empirical evaluation. Finally, we show through experiments that our theoretical bound qualitatively mirrors the actual performance as a function of the step size, even for nonconvex problems.

Problem

• step-size stability
• stochastic optimization
• adaptive step-size
• suboptimality bound
• SGD

Innovation

• step-size stability
• stochastic optimization
• adaptive step-size methods
• theoretical analysis
• robustness