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On Safety in Safe Bayesian Optimization

Christian Fiedler, Johanna Menn, Lukas Kreisköther, Sebastian Trimpe

TL;DR

The Lipschitz-only Safe Bayesian Optimization (LoSBO) algorithm is introduced, which guarantees safety without an assumption on the RKHS bound, and empirically show that this algorithm is not only safe, but also exhibits superior performance compared to the state-of-the-art on several function classes.

Abstract

Optimizing an unknown function under safety constraints is a central task in robotics, biomedical engineering, and many other disciplines, and increasingly safe Bayesian Optimization (BO) is used for this. Due to the safety critical nature of these applications, it is of utmost importance that theoretical safety guarantees for these algorithms translate into the real world. In this work, we investigate three safety-related issues of the popular class of SafeOpt-type algorithms. First, these algorithms critically rely on frequentist uncertainty bounds for Gaussian Process (GP) regression, but concrete implementations typically utilize heuristics that invalidate all safety guarantees. We provide a detailed analysis of this problem and introduce Real-\b{eta}-SafeOpt, a variant of the SafeOpt algorithm that leverages recent GP bounds and thus retains all theoretical guarantees. Second, we identify assuming an upper bound on the reproducing kernel Hilbert space (RKHS) norm of the target function, a key technical assumption in SafeOpt-like algorithms, as a central obstacle to real-world usage. To overcome this challenge, we introduce the Lipschitz-only Safe Bayesian Optimization (LoSBO) algorithm, which guarantees safety without an assumption on the RKHS bound, and empirically show that this algorithm is not only safe, but also exhibits superior performance compared to the state-of-the-art on several function classes. Third, SafeOpt and derived algorithms rely on a discrete search space, making them difficult to apply to higher-dimensional problems. To widen the applicability of these algorithms, we introduce Lipschitz-only GP-UCB (LoS-GP-UCB), a variant of LoSBO applicable to moderately high-dimensional problems, while retaining safety.

On Safety in Safe Bayesian Optimization

TL;DR

The Lipschitz-only Safe Bayesian Optimization (LoSBO) algorithm is introduced, which guarantees safety without an assumption on the RKHS bound, and empirically show that this algorithm is not only safe, but also exhibits superior performance compared to the state-of-the-art on several function classes.

Abstract

Optimizing an unknown function under safety constraints is a central task in robotics, biomedical engineering, and many other disciplines, and increasingly safe Bayesian Optimization (BO) is used for this. Due to the safety critical nature of these applications, it is of utmost importance that theoretical safety guarantees for these algorithms translate into the real world. In this work, we investigate three safety-related issues of the popular class of SafeOpt-type algorithms. First, these algorithms critically rely on frequentist uncertainty bounds for Gaussian Process (GP) regression, but concrete implementations typically utilize heuristics that invalidate all safety guarantees. We provide a detailed analysis of this problem and introduce Real-\b{eta}-SafeOpt, a variant of the SafeOpt algorithm that leverages recent GP bounds and thus retains all theoretical guarantees. Second, we identify assuming an upper bound on the reproducing kernel Hilbert space (RKHS) norm of the target function, a key technical assumption in SafeOpt-like algorithms, as a central obstacle to real-world usage. To overcome this challenge, we introduce the Lipschitz-only Safe Bayesian Optimization (LoSBO) algorithm, which guarantees safety without an assumption on the RKHS bound, and empirically show that this algorithm is not only safe, but also exhibits superior performance compared to the state-of-the-art on several function classes. Third, SafeOpt and derived algorithms rely on a discrete search space, making them difficult to apply to higher-dimensional problems. To widen the applicability of these algorithms, we introduce Lipschitz-only GP-UCB (LoS-GP-UCB), a variant of LoSBO applicable to moderately high-dimensional problems, while retaining safety.
Paper Structure (24 sections, 1 theorem, 20 equations, 8 figures, 1 table, 4 algorithms)

This paper contains 24 sections, 1 theorem, 20 equations, 8 figures, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Outline
  3. Background
  4. Gaussian Processes
  5. Reproducing kernel Hilbert spaces
  6. Frequentist Uncertainty Bounds
  7. Problem setting and objectives
  8. Related work
  9. Safety in Bayesian Optimization
  10. Lipschitz-based methods
  11. Frequentist uncertainty bounds and practical safety issues in SafeOpt
  12. Practical safety issues in SafeOpt
  13. Real-$\beta$-SafeOpt
  14. Lipschitz-only Safe Bayesian Optimization (LoSBO)
  15. Practical problems with the RKHS norm bound
  16. ...and 9 more sections

Key Result

Proposition 1

Let $f: D \rightarrow \mathbb{R}$ be an $L$-Lipschitz function. Assume that $|\epsilon_t| \leq E$ for all $t \geq 1$ and let $\emptyset \not = S_0 \subseteq D$ such that $f(x) \geq h$ for all $x \in S_0$. For any choice of the scaling factors $\beta_t>0$, running the LoSBO algorithm leads to a seque

Figures (8)

  • Figure 1: Illustration of the required GP error bounds. Consider a fixed ground truth (solid black line), of which only finitely many samples are known (blue crosses). Applying GP regression leads to a posterior GP, from which a high-probability uncertainty set can be derived (shaded blue). Left: The ground truth is completely contained in the uncertainty set. Right: The ground truth violates the uncertainty bound around $x=1$.
  • Figure 2: Illustration of SafeOpt. The safe set (gray bar), expanders (green bar), and maximizers (blue bar), which are derived from the current GP model (solid blue line is the posterior mean, shaded blue area are the uncertainty sets), are used to find the safely reachable optimum (red box). In each iteration, the next input is chosen from the union of the current expanders and maximizers (a subset of the safe set) by maximizing the acquisition function.
  • Figure 3: Illustration of LosBO being safe while a safety definition in based on leads to unsafe points in the safe set. The safe set of losbo (gray set) is determined by the constant $E$ (gray arrow) and the Lipschitz cone (orange). The GP mean and the confidence bounds are illustrated in blue. The points in the safe set given by the lower confidence bound are green if they are safe and red if they are unsafe.
  • Figure 4: Comparison of LosBO and Real-$\beta$-SafeOpt in a well-specified setting. Thick solid lines are the means over all functions and repetitions, thin solid lines are the means over all repetitions for each individual function, shaded area corresponds to one standard deviation over all runs.
  • Figure 5: Comparison of LosBO and Real-$\beta$-SafeOpt in misspecified settings.
  • ...and 3 more figures

Theorems & Definitions (5)

  • Remark 1
  • Proposition 1
  • proof
  • Remark 2
  • Remark 3