From the learning theory perspective, the learning guarantees of SCO algorithms with known derivatives have been studied in the literature.

Stochastic compositional optimization (SCO) problem constitutes a class of optimization problems characterized by the objective function with a compositional form, including the tasks with known derivatives, such as AUC maximization, and the derivative-free tasks exemplified by black-box vertical federated learning (VFL). From the learning theory perspective, the learning guarantees of SCO algorithms with known derivatives have been studied in the literature. However, the potential impacts of the derivative-free setting on the learning guarantees of SCO remains unclear and merits further investigation. This paper aims to reveal the impacts by developing a theoretical analysis for two derivative-free algorithms, black-box SCGD and SCSC. Specifically, we first provide the sharper generalization upper bounds of convex SCGD and SCSC based on a new stability analysis framework more effective than prior work under some milder conditions, which is further developed to the non-convex case using the almost co-coercivity property of smooth function. Then, we derive the learning guarantees of three black-box variants of non-convex SCGD and SCSC with additional optimization analysis. Comparing these results, we theoretically uncover the impacts that a better gradient estimation brings a tighter learning guarantee and a larger proportion of unknown gradients may lead to a stronger dependence on the gradient estimation quality. Finally, our analysis is applied to two SCO algorithms, FOO-based vertical VFL and VFL-CZOFO, to build the first learning guarantees for VFL that align with the findings of SCGD and SCSC.

From the learning theory perspective, the learning guarantees of SCO algorithms with known derivatives have been studied in the literature.

Stochastic compositional optimization (SCO) problem constitutes a class of optimization problems characterized by the objective function with a compositional form, including the tasks with known derivatives, such as AUC maximization, and the derivative-free tasks exemplified by black-box vertical federated learning (VFL). From the learning theory perspective, the learning guarantees of SCO algorithms with known derivatives have been studied in the literature. However, the potential impacts of the derivative-free setting on the learning guarantees of SCO remains unclear and merits further investigation. This paper aims to reveal the impacts by developing a theoretical analysis for two derivative-free algorithms, black-box SCGD and SCSC. Specifically, we first provide the sharper generalization upper bounds of convex SCGD and SCSC based on a new stability analysis framework more effective than prior work under some milder conditions, which is further developed to the non-convex case using the almost co-coercivity property of smooth function.

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.