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Bayesian distributionally robust variational inequalities: regularization and quantification

Wentao Ma, Zhiping Chen, Xiaojun Chen

Abstract

We propose a Bayesian distributionally robust variational inequality (DRVI) framework that models the data-generating distribution through a finite mixture family, which allows us to study the DRVI on a tractable finite-dimensional parametric ambiguity set. To address distributional uncertainty, we construct a data-driven ambiguity set with posterior coverage guarantees via Bayesian inference. We also employ a regularization approach to ensure numerical stability. We prove the existence of solutions to the Bayesian DRVI and the asymptotic convergence to a solution as sample size grows to infinity and the regularization parameter goes to zero. Moreover, we derive quantitative stability bounds and finite-sample guarantees under data scarcity and contamination. Numerical experiments on a distributionally robust multi-portfolio Nash equilibrium problem validate our theoretical results and demonstrate the robustness and reliability of Bayesian DRVI solutions in practice.

Bayesian distributionally robust variational inequalities: regularization and quantification

Abstract

We propose a Bayesian distributionally robust variational inequality (DRVI) framework that models the data-generating distribution through a finite mixture family, which allows us to study the DRVI on a tractable finite-dimensional parametric ambiguity set. To address distributional uncertainty, we construct a data-driven ambiguity set with posterior coverage guarantees via Bayesian inference. We also employ a regularization approach to ensure numerical stability. We prove the existence of solutions to the Bayesian DRVI and the asymptotic convergence to a solution as sample size grows to infinity and the regularization parameter goes to zero. Moreover, we derive quantitative stability bounds and finite-sample guarantees under data scarcity and contamination. Numerical experiments on a distributionally robust multi-portfolio Nash equilibrium problem validate our theoretical results and demonstrate the robustness and reliability of Bayesian DRVI solutions in practice.

Paper Structure

This paper contains 10 sections, 23 theorems, 112 equations, 2 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Model setup and background
  3. Bayesian parametric ambiguity sets for distributions
  4. Model component estimation and comparison with classical ambiguity sets
  5. Existence and convergence analysis of solutions
  6. Existence of solutions for \ref{['bdrvi2']} and \ref{['bdrvi-regular2']}
  7. Convergence analysis
  8. Stability of solution sets
  9. Numerical experiment
  10. Concluding remarks.

Key Result

Lemma 1

Under Assumptions ass-bvm-ass3.1, $\mathbb{H}({\Theta}_N,\{\theta^c\})$ converges to 0 almost surely as $N\to\infty$.

Figures (2)

  • Figure 1: CDF envelopes induced by different ambiguity sets at $\alpha=0.05$.
  • Figure 2: The relationship diagram of problems \ref{['bdrvi-true']}, \ref{['bdrvi2']}, \ref{['bdrvi-regular2']}, \ref{['p-bdrvi2']}.

Theorems & Definitions (32)

  • Example 1
  • Definition 1: Frequentist guarantee
  • Definition 2: Posterior guarantee
  • Lemma 1
  • Theorem 1
  • Proposition 1
  • Theorem 2
  • Remark 1
  • Lemma 2
  • Lemma 3
  • ...and 22 more