Step-Size Stability in Stochastic Optimization: A Theoretical Perspective

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 non-convex problems.

When training machine learning models, one important configuration is to set the step size/learning rate of the training algorithm correctly. Algorithms that perform well for a wider range of learning rates are usually preferable. Previous work presented several optimization algorithms that are better than the classical stochastic gradient descent in this regard (empirically). Our work gives a theoretical argument for this phenomenon by identifying a key quantity in the theoretical analysis. This is an important step in order to understand/explain why certain algorithms are naturally more stable than others.