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We propose a statistically optimal approach to construct data-drivendecisions for stochastic optimization problems. Fundamentally, a data-drivendecision is simply a function that maps the available training data to afeasible action. It can always be expressed as the minimizer of a surrogateoptimization model constructed from the data. The quality of a data-drivendecision is measured by its out-of-sample risk. An additional quality measureis its out-of-sample disappointment, which we define as the probability thatthe out-of-sample risk exceeds the optimal value of the surrogate optimizationmodel. An ideal data-driven decision should minimize the out-of-sample risksimultaneously with respect to every conceivable probability measure as thetrue measure is unkown. Unfortunately, such ideal data-driven decisions aregenerally unavailable. This prompts us to seek data-driven decisions thatminimize the in-sample risk subject to an upper bound on the out-of-sampledisappointment. We prove that such Pareto-dominant data-driven decisions existunder conditions that allow for interesting applications: the unknowndata-generating probability measure must belong to a parametric ambiguity set,and the corresponding parameters must admit a sufficient statistic thatsatisfies a large deviation principle. We can further prove that the surrogateoptimization model must be a distributionally robust optimization problemconstructed from the sufficient statistic and the rate function of its largedeviation principle. Hence the optimal method for mapping data to decisions isto solve a distributionally robust optimization model. Maybe surprisingly, thisresult holds even when the training data is non-i.i.d. Our analysis reveals howthe structural properties of the data-generating stochastic process impact theshape of the ambiguity set underlying the optimal distributionally robustmodel.

A Pareto Dominance Principle for Data-Driven Optimization