Exactly or Approximately Wasserstein Distributionally Robust Estimation According to Wasserstein Radii Being Small or Large

TL;DR

The paper studies robust estimation under joint Wasserstein-distance uncertainty for the parameter and noise distributions in a linear model, formulating an infinite-dimensional nonconvex minimax WDRE problem. It establishes that a WDRE saddle point exists iff a corresponding finite-dimensional (Gaussian, linear) problem has a saddle point, but also provides a counterexample showing nonexistence in general, motivating verifiable conditions. A verifiable necessary-and-sufficient condition and a simple sufficient condition are derived, with the latter ensuring existence when Wasserstein radii are small; in the absence of a saddle point, a tight convex SDP relaxation yields an upper bound and an explicit robust linear estimator constructed from primal–dual solutions. Numerical simulations validate the theory, illustrating nonexistence scenarios, the conservative yet informative nature of the sufficient condition, and robustness gains of the proposed estimator in higher dimensions.

Abstract

This paper primarily considers the robust estimation problem under Wasserstein distance constraints on the parameter and noise distributions in the linear measurement model with additive noise, which can be formulated as an infinite-dimensional nonconvex minimax problem. We prove that the existence of a saddle point for this problem is equivalent to that for a finite-dimensional minimax problem, and give a counterexample demonstrating that the saddle point may not exist. Motivated by this observation, we present a verifiable necessary and sufficient condition whose parameters can be derived from a convex problem and its dual. Additionally, we also introduce a simplified sufficient condition, which intuitively indicates that when the Wasserstein radii are small enough, the saddle point always exists. In the absence of the saddle point, we solve an finite-dimensional nonconvex minimax problem, obtained by restricting the estimator to be linear. Its optimal value establishes an upper bound on the robust estimation problem, while its optimal solution yields a robust linear estimator. Numerical experiments are also provided to validate our theoretical results.

Figures (3)

• Figure 5.1: MSE vs $\rho_w$ when $\rho_x=3$ for $d=20$.
• Figure 5.2: The actual bound and the lower bound for the existence of the saddle point.
• Figure 5.3: The frequency histograms of the RMSE.

Theorems & Definitions (22)

• Lemma 3.1
• Lemma 3.2
• proof
• Remark
• Theorem 3.3
• proof
• Remark
• Lemma 3.4
• Lemma 3.5
• proof