Towards Safe Multi-Task Bayesian Optimization (2312.07281v3)

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.

• The paper presents a robust Bayesian optimization method that safely extends error bounds to accommodate unknown hyperparameters in multi-task settings.
• It integrates multi-task Gaussian Processes to efficiently leverage supplementary data, thereby reducing the need for costly direct evaluations.
• Simulation results using laser synchronization models show that SaMSBO outperforms state-of-the-art methods in both safety and evaluation efficiency.

Introduction to Bayesian Optimization

Bayesian optimization (BO) is a method for the efficient optimization of expensive-to-evaluate functions. It builds a probabilistic model, typically using Gaussian Processes (GP), to predict the function's behavior and suggest the most promising points to evaluate next. The method is particularly suited for scenarios with noisy and costly function evaluations. Multi-task Gaussian Processes extend this concept, allowing the use of correlated information from related tasks to further improve optimization efficiency.

The Challenge of Safe Optimization

A critical issue in online optimization, particularly for physical systems, is safety. Evaluations that could harm the system must be avoided. A common approach to ensuring safety in BO is to use error bounds to exclude unsafe inputs from consideration, but this presumes certain known hyperparameters defining the GP model. Given that these parameters are commonly not known in advance in practice, the researchers have developed robust bounds within a Bayesian framework that accommodate unknown hyperparameters. This work extends these robust bounds to apply them in a multi-task setting.

The Multi-Task Setting

In multi-task Bayesian optimization, the goal is to learn about the main task using not only direct observations but also correlated models or 'supplementary tasks'. These models are less expensive to evaluate and provide additional information about the main task. When combined, assessments from both the main task and supplementary models can substantially accelerate optimization. This paper presents a mechanism for performing robust and safe BO in such a multi-task context, handling uncertainty in the correlation between tasks.

Proposed Algorithm and Simulation Results

The paper's primary contribution is a Bayesian optimization algorithm that considers safety and leverages multi-task GP models to predict the performance of various tasks. By extending the robust error bounds to incorporate uncertain correlation matrices between tasks, the algorithm ensures that optimization is performed safely. Simulations using models from the European XFEL's laser-based synchronization system demonstrate that the proposed algorithm, SaMSBO, not only maintains safety but also outperforms other state-of-the-art methods in terms of solution quality and efficiency in function evaluations.

The algorithm's computational complexity, particularly in later stages due to Markov chain Monte Carlo methods, is a noted trade-off. The increased per-iteration cost is offset by the overall reduction in the number of function evaluations required. The researchers aim to enhance exploration strategies in future work and test the algorithm in real-world applications.