Sample Complexity of Sample Average Approximation for Conditional Stochastic Optimization

Abstract

In this paper, we study a class of stochastic optimization problems, referred to as the Conditional Stochastic Optimization (CSO), in the form of min x∈X E ξ f ξ E η|ξ [g η (x, ξ)] . CSO finds a wide spectrum of applications including portfolio selection, reinforcement learning, robust and invariant learning. We establish the sample complexity of the sample average approximation (SAA) for CSO, under a variety of structural assumptions, such as Lipschitz continuity, smoothness, and error bound conditions. We show that the total sample complexity improves from O(d/ 4 ) to O(d/ 3 ) when assuming smoothness of the outer function, and further to O(1/ 2 ) when the empirical function satisfies the quadratic growth condition. We also establish the sample complexity of a modified SAA, when ξ and η are independent. Our numerical results from several experiments further support our theoretical findings.

Biased Stochastic Gradient Descent for Conditional Stochastic Optimization

“…This improves to O( −2 ) if both conditions hold, where we use O(•) to represent the rate hiding the logarithmic factors. In contrast to the SAA results in Hu et al [2019], these sample complexities are independent of the problem's dimensions. Furthermore, we show that, for weakly convex CSO problems (which are not necessarily smooth nor convex), BSGD requires a total sample complexity of O( −8 ) to achieve an -stationary point.…”

“…The above result indicates that smoothness conditions make a difference in the total sample complexity of BSGD when solving CSO. It is worth pointing out that the sample complexity of BSGD matches with the that of ERM for strongly convex objectives esatblished in Hu et al [2019].…”

“…However, this approach requires convexity of f and linearity of g, which are not satisfied in general applications especially when neural networks are used. SAA [Hu et al, 2019]…”

“…In this paper, we focus on the general CSO problem where multiple samples from the conditional distribution η|ξ are available, and the objective is not necessarily in the compositional form of a convex loss f ξ (•) and a linear mapping g η (•, ξ). A closely related work is Hu et al [2019], in which the authors study the generalization error bound and sample complexity of empirical risk minimization (ERM), a.k.a., sample average approximation (SAA) for general CSO. Differently, here we aim at developing efficient stochastic gradient-based methods that directly solve the CSO problem (1), for both convex and nonconvex settings.…”

“…For convex problems, this refers to the sample complexity required to find an -optimal solution in expectation; for nonconvex problems, this refers to the sample complexity required to find an -stationary point in expectation. Some of our results are summarized in Table 1 with a comparison to the sample complexity of SAA (or ERM) established in Hu et al [2019].…”