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When the underlying probability distribution in a stochastic optimization isobserved only through data, various data-driven formulations have been studiedto obtain approximate optimal solutions. We show that no such formulations can,in a sense, theoretically improve the statistical quality of the solutionobtained from empirical optimization. We argue this by proving that thefirst-order behavior of the optimality gap against the oracle best solution,which includes both the bias and variance, for any data-driven solution issecond-order stochastically dominated by empirical optimization, as long assuitable smoothness holds with respect to the underlying distribution. Wedemonstrate this impossibility of improvement in a range of examples includingregularized optimization, distributionally robust optimization, parametricoptimization and Bayesian generalizations. We also discuss the connections ofour results to semiparametric statistical inference and other perspectives inthe data-driven optimization literature.

On the Impossibility of Statistically Improving Empirical Optimization: A Second-Order Stochastic Dominance Perspective