Distributionally-Robust Optimization with Noisy Data for Discrete Uncertainties Using Total Variation Distance

TL;DR

This paper develops a distributionally-robust optimization framework for discrete uncertainties with noisy data, using total variation distance to ensure asymptotic consistency and tractability in large-scale stochastic problems.

Contribution

It introduces a novel ambiguity set capturing noisy data via total variation distance, ensuring asymptotic convergence and convex reformulation for robust optimization.

Findings

Solutions converge to original stochastic programs as data grows

The approach is tractable and suitable for large-scale problems

Proven under the assumption of uniformly diagonally dominant noise distribution

Abstract

Stochastic programs where the uncertainty distribution must be inferred from noisy data samples are considered. The stochastic programs are approximated with distributionally-robust optimizations that minimize the worst-case expected cost over ambiguity sets, i.e., sets of distributions that are sufficiently compatible with the observed data. In this paper, the ambiguity sets capture the set of probability distributions whose convolution with the noise distribution remains within a ball centered at the empirical noisy distribution of data samples parameterized by the total variation distance. Using the prescribed ambiguity set, the solutions of the distributionally-robust optimizations converge to the solutions of the original stochastic programs when the numbers of the data samples grow to infinity. Therefore, the proposed distributionally-robust optimization problems are asymptotically consistent. This is proved under the assumption that the distribution of the noise is uniformly diagonally dominant. More importantly, the distributionally-robust optimization problems can be cast as tractable convex optimization problems and are therefore amenable to large-scale stochastic problems.