Don't Look Back in Anger: Wasserstein Distributionally Robust Optimization with Nonstationary Data

Abstract: We study data-driven decision problems where historical observations are generated by a time-evolving distribution whose consecutive shifts are bounded in Wasserstein distance. We address this nonstationarity using a distributionally robust optimization model with an ambiguity set that is a Wasserstein ball centered at a weighted empirical distribution, thereby allowing for the time decay of past data in a way which accounts for the drift of the data-generating distribution. Our main technical contribution is a concentration inequality for weighted empirical distributions that explicitly captures both the effective sample size (i.e., the equivalent number of equally weighted observations) and the distributional drift. Using our concentration inequality, we select observation weights that optimally balance the effective sample size against the extent of drift. The family of optimal weightings reveals an interplay between the order of the Wasserstein ambiguity ball and the time-decay profile of the optimal weights. Classical weighting schemes, such as time windowing and exponential smoothing, emerge as special cases of our framework, for which we derive principled choices of the parameters. Numerical experiments demonstrate the effectiveness of the proposed approach.