An Analysis of Safety Guarantees in Multi-Task Bayesian Optimization

TL;DR

This work tackles safe optimization of expensive black-box objectives by integrating auxiliary information through multi-task Gaussian processes. It develops SaMSBO, a safe Bayesian optimization method that derives both frequentist and Bayesian uniform error bounds under uncertain task correlations, using a hyper-posterior over the correlation matrix and a confidence set $\mathcal{C}_\rho$. The key contributions are the extension of multi-task scaling factors to safety guarantees, improvements to single-task Bayesian bounds, and empirical evidence on Powell, Branin, and a control problem showing reduced main-task evaluations while preserving safety. The approach enhances sample efficiency in high-cost settings by leveraging related, cheaper auxiliary information during optimization.

Abstract

This paper addresses the integration of additional information sources into a Bayesian optimization framework while ensuring that safety constraints are satisfied. The interdependencies between these information sources are modeled using an unknown correlation matrix. We explore how uniform error bounds must be adjusted to maintain constraint satisfaction throughout the optimization process, considering both Bayesian and frequentist statistical perspectives. This is achieved by appropriately scaling the error bounds based on a confidence interval that can be estimated from the data. Furthermore, the efficacy of the proposed approach is demonstrated through experiments on two benchmark functions and a controller parameter optimization problem. Our results highlight a significant improvement in sample efficiency, demonstrating the methods suitability for optimizing expensive-to-evaluate functions.

Figures (4)

• Figure 1: Overview of different safe settings with safety threshold $T$ illustrated by "- - -". In (a) the single-task setting $\left(\bar{\beta} = \beta\right)$ is depicted, i. e., no simulation samples are considered, and the safe region is the smallest. (b) shows the multi-task setting with slight correlation and (c) with high correlation. In both cases (b) and (c), using information from an additional task increases the safe region.
• Figure 2: Comparison of the proposed multi-task safe algorithm SaMSBO with the single-task equivalence Safe-UCBSui2015 on the Powell and Branin function. The abscissa denotes the number of evaluations of the main task and the ordinates the best observation. The shaded area represents the standard deviation.
• Figure 3: Illustration of the interconnected system. The blocks $F_r$ and $F_i, i=1,\dots,N$ denote disturbance filters which colorize the white noise inputs $w_j, j=1,\dots,N+1$. $G_i$ denote the laser plants and $K_i$ PI controllers for each subsystem.
• Figure 4: Comparison of the proposed multi-task safe algorithm SaMSBO with its single-task equivalent Safe-UCBSui2015 and none-safe single- and multi-task algorithms. The multi-task algorithms have access to one supplementary task which is disturbed by 0.3. The abscissa denotes the number of evaluations of the main task and the ordinate the best cost value. The lines and dashed lines denote the average and the shaded area represents the standard deviation.

Theorems & Definitions (10)

• Lemma 3
• Lemma 4
• Lemma 5
• Proposition 6
• Definition 8
• Proposition 9
• Lemma 11
• Theorem 12
• Lemma 13
• Corollary 14