Stability and Generalization of Stochastic Compositional Gradient Descent Algorithms
This work addresses a theoretical gap for researchers in machine learning, particularly in fields like reinforcement learning and meta-learning, by offering foundational stability and generalization guarantees for SCO algorithms, though it is incremental as it builds on existing optimization frameworks.
The paper tackles the lack of generalization analysis for stochastic compositional gradient descent algorithms by introducing compositional uniform stability and deriving dimension-independent excess risk bounds for SCGD and SCSC algorithms, providing the first-known results in this area.
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.