Sample Average Approximation for Distributionally Robust Optimization with $\phi$-divergences

Abstract

It is well known that estimating the expectation of any given bounded random variable with values in $[-B, B]$ has two basic properties: (1) the number of samples needed scales at the order of $O(B^2/\epsilon^2)$, where $\epsilon$ is the prescribed target precision, and (2) such a sample complexity is independent of the underlying probability measure. We show that neither of these properties can no longer hold when evaluating the worst-case expectation of the random variable, where the probability measures defining the expectation belong to a $\phi$-divergence ball centered at some nominal measure $P$. Specifically, the sample complexity and its dependence on the nominal measure can be completely characterized by the growth of the divergence function. When the divergence function $\phi$ exhibits superlinear growth, a $P$-independent sample complexity can be obtained for sample average approximation, which depends only on the growth of $\phi$, the radius of the divergence ball, and the target precision. We also provide sample complexity lower bounds and demonstrate the optimality of the obtained bounds for commonly used $\phi$-divergences. On the other hand, when superlinear growth does not hold for $\phi$, we show that for any estimation method, evaluating the worst-case expectation has a $P$-dependent sample complexity lower bound that can be made arbitrarily large by changing $P$. In this case, we discuss the approach of adopting the generalized definition of $\phi$-divergence, through which $P$-independent sample complexity can be recovered, and hence demonstrate the essential role of absolute continuity in affecting statistical tractability.