Distributionally robust risk evaluation with an isotonic constraint

Abstract Statistical learning under distribution shift is challenging when neither prior knowledge nor data from the target distribution is available. Distributionally robust learning (DRL) aims to control the worst-case statistical performance within a set of candidate distributions, but how to properly specify the set remains challenging. To enable distributional robustness without being overly conservative, in this paper we propose a shape-constrained approach to DRL, which incorporates prior information about the way in which the unknown target distribution differs from its estimate—specifically, we assume the unknown density ratio between the target distribution and its estimate is isotonic with respect to some partial order. At the population level, we provide a solution to the shape-constrained optimization problem that can be solved without the challenge of an explicit isotonic constraint. At the sample level, we provide consistency results for an empirical estimator of the target in a range of different settings. Empirical studies on both synthetic and real data demonstrate the improved efficiency of the proposed shape-constrained approach.