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Wasserstein Distributionally Robust Bayesian Optimization with Continuous Context

Francesco Micheli, Efe C. Balta, Anastasios Tsiamis, John Lygeros

TL;DR

The paper tackles sequential decision-making under uncertain, continuously distributed contexts by introducing Wasserstein Distributionally Robust Bayesian Optimization (WDRBO). It develops a tractable Lipschitz-based reformulation that avoids context discretization and provides two settings: General WDRBO with a specified ambiguity set and Data-Driven WDRBO built from past observations, both achieving sublinear regret; the data-driven variant leverages finite-sample Wasserstein concentration to avoid strict decay assumptions on the ambiguity radius. The authors establish regret bounds that match state-of-the-art results in RKHS-based BO, and show empirical robustness and computational efficiency across synthetic and real-world tasks. This work broadens robust optimization in BO to continuous-context scenarios with practical guarantees and scalable performance, enabling reliable optimization under distributional context shifts.

Abstract

We address the challenge of sequential data-driven decision-making under context distributional uncertainty. This problem arises in numerous real-world scenarios where the learner optimizes black-box objective functions in the presence of uncontrollable contextual variables. We consider the setting where the context distribution is uncertain but known to lie within an ambiguity set defined as a ball in the Wasserstein distance. We propose a novel algorithm for Wasserstein Distributionally Robust Bayesian Optimization that can handle continuous context distributions while maintaining computational tractability. Our theoretical analysis combines recent results in self-normalized concentration in Hilbert spaces and finite-sample bounds for distributionally robust optimization to establish sublinear regret bounds that match state-of-the-art results. Through extensive comparisons with existing approaches on both synthetic and real-world problems, we demonstrate the simplicity, effectiveness, and practical applicability of our proposed method.

Wasserstein Distributionally Robust Bayesian Optimization with Continuous Context

TL;DR

The paper tackles sequential decision-making under uncertain, continuously distributed contexts by introducing Wasserstein Distributionally Robust Bayesian Optimization (WDRBO). It develops a tractable Lipschitz-based reformulation that avoids context discretization and provides two settings: General WDRBO with a specified ambiguity set and Data-Driven WDRBO built from past observations, both achieving sublinear regret; the data-driven variant leverages finite-sample Wasserstein concentration to avoid strict decay assumptions on the ambiguity radius. The authors establish regret bounds that match state-of-the-art results in RKHS-based BO, and show empirical robustness and computational efficiency across synthetic and real-world tasks. This work broadens robust optimization in BO to continuous-context scenarios with practical guarantees and scalable performance, enabling reliable optimization under distributional context shifts.

Abstract

We address the challenge of sequential data-driven decision-making under context distributional uncertainty. This problem arises in numerous real-world scenarios where the learner optimizes black-box objective functions in the presence of uncontrollable contextual variables. We consider the setting where the context distribution is uncertain but known to lie within an ambiguity set defined as a ball in the Wasserstein distance. We propose a novel algorithm for Wasserstein Distributionally Robust Bayesian Optimization that can handle continuous context distributions while maintaining computational tractability. Our theoretical analysis combines recent results in self-normalized concentration in Hilbert spaces and finite-sample bounds for distributionally robust optimization to establish sublinear regret bounds that match state-of-the-art results. Through extensive comparisons with existing approaches on both synthetic and real-world problems, we demonstrate the simplicity, effectiveness, and practical applicability of our proposed method.

Paper Structure

This paper contains 15 sections, 105 equations, 4 figures, 2 tables, 1 algorithm.

Table of Contents

  1. INTRODUCTION
  2. RELATED WORK
  3. PROBLEM FORMULATION
  4. Wasserstein Distributionally Robust Objective
  5. Regularity Assumptions and Surrogate Model
  6. WASSERSTEIN DISTRIBUTIONALLY ROBUST BAYESIAN OPTIMIZATION
  7. General WDRBO
  8. Data-Driven WDRBO
  9. EXPERIMENTS
  10. CONCLUSIONS AND FUTURE WORK
  11. BACKGROUND ON RKHS AND KERNEL RIDGE REGRESSION
  12. Kernels that satisfy the Lipschitz condition in Assumption 1
  13. MAIN PROOFS
  14. Proofs of the data-driven setting
  15. EXPERIMENTS DETAILS AND ADDITIONAL EXPERIMENTS

Figures (4)

  • Figure 1: (a) Function $f(x,c)$ and its optima under optima under $\hat{\mathcal{P}}_t$ and $\mathcal{P}^*_t$. (b) Mean and standard error of the cumulative expected regret.
  • Figure 2: Mean and standard error of the cumulative expected regret.
  • Figure 3: Mean and standard error of the cumulative expected regret.
  • Figure 4: Mean and standard error of the instantaneous expected regret.

Theorems & Definitions (11)

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  • ...and 1 more