Stability and Generalization of Stochastic Compositional Gradient Descent Algorithms

TL;DR

This paper analyzes the stability and generalization properties of stochastic compositional gradient descent algorithms, providing the first such results and establishing their implications for machine learning tasks involving nested expectations.

Contribution

It introduces the concept of compositional uniform stability and derives the first stability and generalization bounds for SCO algorithms like SCGD and SCSC.

Findings

Established compositional uniform stability for SCGD and SCSC.

Derived dimension-independent excess risk bounds for these algorithms.

Linked stability results to generalization performance in SCO problems.

Abstract

Many machine learning tasks can be formulated as a stochastic compositional optimization (SCO) problem such as reinforcement learning, AUC maximization, and meta-learning, where the objective function involves a nested composition associated with an expectation. While a significant amount of studies has been devoted to studying the convergence behavior of SCO algorithms, there is little work on understanding their generalization, i.e., how these learning algorithms built from training examples would behave on future test examples. In this paper, we provide the stability and generalization analysis of stochastic compositional gradient descent algorithms through the lens of algorithmic stability in the framework of statistical learning theory. Firstly, we introduce a stability concept called compositional uniform stability and establish its quantitative relation with generalization for SCO problems. Then, we establish the compositional uniform stability results for two popular stochastic compositional gradient descent algorithms, namely SCGD and SCSC. Finally, we derive dimension-independent excess risk bounds for SCGD and SCSC by trade-offing their stability results and optimization errors. To the best of our knowledge, these are the first-ever-known results on stability and generalization analysis of stochastic compositional gradient descent algorithms.