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.

翻译：许多机器学习任务（如强化学习、AUC最大化及元学习）可被建模为随机组合优化问题，其目标函数涉及与期望相关的嵌套组合。尽管已有大量研究致力于探索随机组合优化算法的收敛行为，但关于其泛化性的研究却相对匮乏——即基于训练样本构建的学习算法在未来测试样本上的表现。本文通过统计学习理论框架下的算法稳定性视角，对随机组合梯度下降算法的稳定性与泛化性展开分析。首先，我们提出一种名为“组合均匀稳定性”的稳定性概念，并建立其与随机组合优化问题泛化性的定量关系。随后，针对两种流行的随机组合梯度下降算法SCGD与SCSC，建立了相应的组合均匀稳定性结果。最终，通过权衡稳定性结果与优化误差，推导出SCGD与SCSC的无维度界超额风险。据我们所知，这是首次对随机组合梯度下降算法进行稳定性与泛化性分析的研究成果。