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

TL;DR

This paper develops a Wasserstein distributionally robust optimization framework for nonstationary data, incorporating time decay and distributional drift, with theoretical concentration bounds and practical weighting schemes.

Contribution

It introduces a concentration bound for weighted empirical distributions under nonstationarity and derives optimal weighting strategies balancing variance and drift.

Findings

Effective handling of nonstationary data with Wasserstein DRO.

Derivation of principled parameter choices for classical weighting schemes.

Numerical experiments confirm the approach's effectiveness.

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 bound 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 bound, we select observation weights that optimally balance variance, determined by the effective sample size, and drift, induced by the temporal changes in the data-generating process. The family of optimal weightings reveals a polynomial relationship between the order of the Wasserstein ambiguity ball and the time-decay profile of the optimal weights. We further characterize how the ambiguity radius must grow with the distributional drift to guarantee a prescribed confidence level. Classical weighting schemes, such as time windowing and simple exponential smoothing, emerge as special cases of our framework, for which we derive principled choices of parameters. Numerical experiments demonstrate the effectiveness of our proposed approach.