Statistics of Robust Optimization: A Generalized Empirical Likelihood Approach
This work addresses robust optimization for researchers in statistics and machine learning, offering incremental improvements in methodology for distributional uncertainty.
The paper tackles statistical inference for stochastic optimization by developing a generalized empirical likelihood framework based on f-divergence balls, providing exact asymptotic coverage for confidence intervals and showing that robustification regularizes problems by their variance.
We study statistical inference and distributionally robust solution methods for stochastic optimization problems, focusing on confidence intervals for optimal values and solutions that achieve exact coverage asymptotically. We develop a generalized empirical likelihood framework---based on distributional uncertainty sets constructed from nonparametric $f$-divergence balls---for Hadamard differentiable functionals, and in particular, stochastic optimization problems. As consequences of this theory, we provide a principled method for choosing the size of distributional uncertainty regions to provide one- and two-sided confidence intervals that achieve exact coverage. We also give an asymptotic expansion for our distributionally robust formulation, showing how robustification regularizes problems by their variance. Finally, we show that optimizers of the distributionally robust formulations we study enjoy (essentially) the same consistency properties as those in classical sample average approximations. Our general approach applies to quickly mixing stationary sequences, including geometrically ergodic Harris recurrent Markov chains.