Sample Complexity of Sample Average Approximation for Conditional Stochastic Optimization
This work addresses theoretical sample complexity bounds for a broad class of stochastic optimization problems used in areas like portfolio selection and reinforcement learning, representing an incremental advance.
The paper tackles the sample complexity of sample average approximation for conditional stochastic optimization, showing improvements from O(d/ε^4) to O(d/ε^3) with smoothness and to O(1/ε^2) with quadratic growth conditions.
In this paper, we study a class of stochastic optimization problems, referred to as the \emph{Conditional Stochastic Optimization} (CSO), in the form of $\min_{x \in \mathcal{X}} \EE_ξf_ξ\Big({\EE_{η|ξ}[g_η(x,ξ)]}\Big)$, which finds a wide spectrum of applications including portfolio selection, reinforcement learning, robust learning, causal inference and so on. Assuming availability of samples from the distribution $\PP(ξ)$ and samples from the conditional distribution $\PP(η|ξ)$, 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 $\cO(d/\eps^4)$ to $\cO(d/\eps^3)$ when assuming smoothness of the outer function, and further to $\cO(1/\eps^2)$ 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. Several numerical experiments further support our theoretical findings. Keywords: stochastic optimization, sample average approximation, large deviations theory