Towards Safe Multi-Task Bayesian Optimization

TL;DR

This paper tackles safe online optimization when multiple information sources with unknown inter-task correlations are available. It extends robust uniform error bounds to a multi-task GP framework by learning a posterior over the correlation matrix $\Sigma$ via MCMC and deriving a robust safety bound $\bar{\beta}$, enabling safe optimization of the main task while exploiting cheaper surrogate models. The authors introduce SaMSBO, the first robustly safe multi-task Bayesian optimization algorithm, and demonstrate its superior performance and sample efficiency compared with state-of-the-art safe BO baselines in simulation. The work significantly reduces evaluation cost for expensive objectives while preserving safety, with practical impact for complex, constrained online systems and simulated surrogate models.

Abstract

Bayesian optimization has emerged as a highly effective tool for the safe online optimization of systems, due to its high sample efficiency and noise robustness. To further enhance its efficiency, reduced physical models of the system can be incorporated into the optimization process, accelerating it. These models are able to offer an approximation of the actual system, and evaluating them is significantly cheaper. The similarity between the model and reality is represented by additional hyperparameters, which are learned within the optimization process. Safety is a crucial criterion for online optimization methods such as Bayesian optimization, which has been addressed by recent works that provide safety guarantees under the assumption of known hyperparameters. In practice, however, this does not apply. Therefore, we extend the robust Gaussian process uniform error bounds to meet the multi-task setting, which involves the calculation of a confidence region from the hyperparameter posterior distribution utilizing Markov chain Monte Carlo methods. Subsequently, the robust safety bounds are employed to facilitate the safe optimization of the system, while incorporating measurements of the models. Simulation results indicate that the optimization can be significantly accelerated for expensive to evaluate functions in comparison to other state-of-the-art safe Bayesian optimization methods, contingent on the fidelity of the models.

Figures (3)

• Figure 1: Overview of different safe Bayesian optimization settings with safety threshold $T$ denoted by "- - -". (a) shows the single-task setting, where no simulation samples are considered, and the safe region is the smallest. (b) visualizes the multi-task setting with slight correlation and (c) with high correlation. In both cases, The multi-task setting increases the safe region.
• Figure 2: 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 3: (a) shows the performance of SaMSBO by tuning a chain for $N = 2$ lasers, where the extra tasks have disturbed filter transfer functions. The line colors indicate the range of the disturbance. (b) shows the performance of different BO algorithms applied on a chain with $N = 5$ lasers. In this trial the filter disturbance lies in the range $\pm10\%$.

Theorems & Definitions (11)

• lemma 1
• proof
• lemma 2
• proof
• theorem 2
• proof
• lemma 3: Schur Product Theorem horn2017
• lemma 4
• proof
• proof : Lemma \ref{['lemma:gamma']}