Step-Size Stability in Stochastic Optimization: A Theoretical Perspective

TL;DR

This paper provides a theoretical analysis of how step size affects stochastic optimization methods, demonstrating that adaptive methods are more robust and aligning theoretical bounds with empirical performance.

Contribution

It introduces a key quantity that measures step-size sensitivity, showing adaptive methods are theoretically more stable than SGD for convex and nonconvex problems.

Findings

Adaptive methods like SPS and NGN are more robust to step size variations.

Theoretical bounds align with empirical performance across problem types.

Step-size sensitivity impacts suboptimality bounds in convex optimization.

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.