Vector Optimization with Gaussian Process Bandits

TL;DR

This paper introduces VOGP, an efficient Gaussian process-based algorithm for black-box vector optimization that leverages preference structures, providing theoretical guarantees and demonstrating significant sample efficiency improvements over existing methods.

Contribution

The paper proposes VOGP, a novel adaptive elimination algorithm for vector optimization with Gaussian processes, offering theoretical guarantees and practical efficiency improvements.

Findings

VOGP achieves approximately 18 times lower sample complexity on average.

Theoretical bounds are derived based on information gain and kernel properties.

Empirical evaluations show VOGP outperforms state-of-the-art algorithms on multiple datasets.

Abstract

We study black-box vector optimization with Gaussian process bandits, where there is an incomplete order relation on objective vectors described by a polyhedral convex cone. Existing black-box vector optimization approaches either suffer from high sample complexity or lack theoretical guarantees. We propose Vector Optimization with Gaussian Process (VOGP), an adaptive elimination algorithm that identifies Pareto optimal solutions sample efficiently by exploiting the smoothness of the objective function. We establish theoretical guarantees, deriving information gain-based and kernel-specific sample complexity bounds. Finally, we conduct a thorough empirical evaluation of VOGP and compare it with the state-of-the-art multi-objective and vector optimization algorithms on several real-world and synthetic datasets, emphasizing VOGP's efficiency (e.g., $\sim18\times$ lower sample complexity on average). We also provide heuristic adaptations of VOGP for cases where the design space is continuous and where the Gaussian process model lacks access to the true kernel hyperparameters. This work opens a new frontier in sample-efficient multi-objective black-box optimization by incorporating preference structures while maintaining theoretical guarantees and practical efficiency.