Stagewise Safe Bayesian Optimization with Gaussian Processes (1806.07555v2)

Abstract: Enforcing safety is a key aspect of many problems pertaining to sequential decision making under uncertainty, which require the decisions made at every step to be both informative of the optimal decision and also safe. For example, we value both efficacy and comfort in medical therapy, and efficiency and safety in robotic control. We consider this problem of optimizing an unknown utility function with absolute feedback or preference feedback subject to unknown safety constraints. We develop an efficient safe Bayesian optimization algorithm, StageOpt, that separates safe region expansion and utility function maximization into two distinct stages. Compared to existing approaches which interleave between expansion and optimization, we show that StageOpt is more efficient and naturally applicable to a broader class of problems. We provide theoretical guarantees for both the satisfaction of safety constraints as well as convergence to the optimal utility value. We evaluate StageOpt on both a variety of synthetic experiments, as well as in clinical practice. We demonstrate that StageOpt is more effective than existing safe optimization approaches, and is able to safely and effectively optimize spinal cord stimulation therapy in our clinical experiments.

• The paper introduces a novel two-stage algorithm that decouples safe region expansion from utility maximization using Gaussian Processes.
• It provides finite-time guarantees for reaching an ε-safe region and converging to an optimal utility value within predefined safety limits.
• Experimental results show faster safe region expansion and higher rewards, demonstrating practical benefits in areas like clinical spinal cord stimulation.

An Examination of "Stagewise Safe Bayesian Optimization with Gaussian Processes"

The paper "Stagewise Safe Bayesian Optimization with Gaussian Processes" introduces a novel approach for sequential decision-making processes where safety is a major concern. The research presented is focused on an efficient algorithm named Stagewise Safe Bayesian Optimization (SSBO), which divides the problem into two distinct stages: safe region expansion and utility function maximization. This methodology aims to optimize unknown utility functions under uncertain safety constraints and is particularly applicable to fields such as medical therapy and robotic control, where both efficacy and safety are paramount.

Overview of the Algorithm and Methodology

Bayesian optimization is leveraged as a powerful and efficient technique for optimizing functions which are expensive to evaluate or lack a closed-form expression. Traditionally, Bayesian optimization interleaves the exploration of the safe region with optimization of the utility function. However, this paper posits that separating these tasks into distinct stages can yield better results. The safe region expansion involves widening the area of the input space that is known to be safe, and once a sufficiently large safe region is established, the focus shifts to maximizing the utility function within this region.

In the proposed SSBO, Gaussian Processes (GPs) are utilized to model both the utility and safety functions. These GPs, defined with zero-mean priors, allow for the quantification of uncertainty via confidence intervals. In the safe expansion stage, these confidence intervals guide the identification of a safe set that can be expanded without violating safety constraints. Once the safe region reaches a defined satisfactory size, the utility maximization stage adopts classical GP-UCB (Gaussian Process Upper Confidence Bound) methods for optimization.

Theoretical Guarantees and Analysis

The paper provides robust theoretical guarantees concerning the algorithm's performance. Two main results are emphasized:

1. Safe Expansion: The paper presents a finite-time guarantee for the algorithm's ability to reach an $\epsilon$-reachable safe region within a certain number of iterations, denoted as $t^*$. The constraints of this safe region are bound by the number of safety functions and their respective Gaussian processes.
2. Utility Optimization: There is also a proven guarantee, within a finite time frame $T_1$, for convergence to an optimal utility value with an accuracy of $\zeta$ within the pre-established safe region. These analytic results are supported through assumptions of bounded RKHS norms and conditions of Lipschitz continuity, which are necessary to maintain such guarantees under the Gaussian process framework.

Experimental Results

Through experimentation on both synthetic problems and a practical clinical trial setting, the SSBO demonstrates superior performance over existing methods, particularly under conditions where safety and utility functions exhibit different domains or scales. In the synthetic scenarios, different configurations of safety constraints and feedback mechanisms are examined to validate the flexibility and effectiveness of SSBO. In comparison to other methods like the constrained Expected Improvement, SSBO showed faster safe region expansion and a higher reward in terms of optimizing the utility.

The practical application on spinal cord stimulation for patients illustrates the algorithm's potential clinical benefits. Here, SSBO helped identify safe and effective therapies more efficiently than expert clinical assessments, underscoring its potential to aid in settings requiring stringent safety compliance.

Implications and Future Directions

The proposed SSBO framework offers substantial improvements and new directions in safe optimization processes. By elegantly decoupling the safe expansion from utility optimization, it provides a more adaptable and potentially faster convergence on optimal solutions that satisfy real-world safety constraints. This decoupling is particularly advantageous in scenarios where safety and utility considerations are significantly disparate.

Future research may delve into extending this framework to dynamic environments or multi-criteria decision-making processes such as those encountered in safe reinforcement learning. Additionally, the exploration of other modeling approaches beyond GPs, such as deep kernel learning, could enhance the flexibility and efficacy of the algorithm in more complex settings or under varied prior conditions.

Overall, this paper offers a comprehensive and innovative approach to addressing safety in Bayesian optimization, with robust theoretical underpinnings and demonstrated efficacy in critical applications.