Wasserstein Distributionally Robust Bayesian Optimization with Continuous Context (2503.20341v1)

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.

• The paper introduces WDRBO, a new algorithm for Bayesian optimization that efficiently handles continuous contextual uncertainty using Wasserstein distance, avoiding previous methods' need for discretization.
• Theoretical analysis establishes finite-sample bounds and sublinear cumulative regret for WDRBO, demonstrating strong performance guarantees even under distributional shifts.
• Empirical results show WDRBO consistently outperforms traditional methods like ERBO, exhibiting scalability and practical potential in areas like logistics and adaptive control.

Wasserstein Distributionally Robust Bayesian Optimization with Continuous Context

Introduction

In the field of sequential data-driven decision-making, particularly under conditions where the distribution of contextual variables is uncertain, this paper presents a noteworthy contribution through a novel algorithm named Wasserstein Distributionally Robust Bayesian Optimization (WDRBO). This algorithm adeptly handles continuous context distributions and is computationally efficient, thereby addressing a critical gap in Bayesian Optimization (BO) and Distributionally Robust Optimization (DRO) literature.

Key Contributions

The paper makes several significant contributions:

1. Algorithm Development: The researchers introduce WDRBO, which blends BO and DRO strategies to optimize sequential decision-making processes. Remarkably, WDRBO allows for continuous context distributions and circumvents the need for context discretization, a limitation inherent in previous DRO approaches.
2. Theoretical Advances: The authors provide a robust theoretical framework grounding WDRBO in finite-sample bounds for distributionally robust optimization and self-normalized concentration in Hilbert spaces. They establish cumulative expected regret bounds of order $\tilde{O}(\sqrt{T}\gamma_T)$ for the general setting and sublinear regret for data-driven contexts, demonstrating its competitive edge over existing methods.
3. Empirical Validation: Through rigorous experimentation on synthetic and real-world datasets, WDRBO is shown to maintain high performance across various scenarios, often with reduced computational complexity compared to other approaches in DRO and BO.

Theoretical Implications

The theoretical implications of this research are critical for expanding the field of Bayesian optimization under uncertainty. By defining the ambiguity set using balls in the Wasserstein distance, the authors lend a flexible and intuitive framework to model context distribution uncertainty, thereby enabling robust decision-making even in environments with substantial distributional shifts. Furthermore, the sublinear regret bounds indicate the method’s efficacy over long sequences of decision-making scenarios, potentially transforming approaches in domains like robotics and scientific experimentation.

Numerical Results

The paper highlights strong numerical results demonstrating algorithm efficacy. Specifically, WDRBO consistently outperforms traditional ERBO (Empirical Risk BO) approaches in cumulative regret, showcasing its robustness to distributional shifts. The experiments underline that WDRBO exhibits promising scalability with contextual dimensionality and maintains a balance between computational overhead and robust optimization capabilities.

Practical Implications and Future Work

Practically, WDRBO offers a scalable solution for industries reliant on sequential decision-making under uncertain conditions, such as logistics optimization and adaptive control in engineering systems. The ability to integrate continuous distributions without losing computational tractability is a substantial advancement.

This work opens new avenues for future research, particularly in adapting the WDRBO framework to include risk-aware metrics like CVaR, and expanding its applicability to dynamic and high-dimensional distributional contexts across various sectors.

In conclusion, the paper enriches Bayesian optimization literature by presenting a comprehensive algorithm capable of handling continuous uncertainty, paving the way for more sophisticated and resilient optimization solutions that accommodate real-world complexities.