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Bayesian Optimization with Formal Safety Guarantees via Online Conformal Prediction

Yunchuan Zhang, Sangwoo Park, Osvaldo Simeone

TL;DR

The paper addresses constrained black-box zero-th order optimization where a safety constraint $q(\mathbf{x})\ge0$ must be respected while maximizing $f(\mathbf{x})$. It introduces SAFE-BOCP, combining Bayesian optimization with online conformal prediction to produce an adaptive, pessimistic safe set that guarantees safety irrespective of the constraint function, at the cost of a non-zero target safety violation rate $\alpha$. Two algorithms are proposed: deterministic D-SAFE-BOCP and probabilistic P-SAFE-BOCP, each with formal safety guarantees (the latter under noise) and a data-driven tail-bound mechanism for practical resilience. The methods are validated on synthetic benchmarks and real-world tasks (e.g., movie recommendations and chemical reactor optimization), demonstrating competitive optimization performance while controlling safety risk and without strong model assumptions about $q$. This work broadens the applicability of safe BO to safety-critical domains by providing assumption-free safety guarantees and practical probability bounds.

Abstract

Black-box zero-th order optimization is a central primitive for applications in fields as diverse as finance, physics, and engineering. In a common formulation of this problem, a designer sequentially attempts candidate solutions, receiving noisy feedback on the value of each attempt from the system. In this paper, we study scenarios in which feedback is also provided on the safety of the attempted solution, and the optimizer is constrained to limit the number of unsafe solutions that are tried throughout the optimization process. Focusing on methods based on Bayesian optimization (BO), prior art has introduced an optimization scheme -- referred to as SAFEOPT -- that is guaranteed not to select any unsafe solution with a controllable probability over feedback noise as long as strict assumptions on the safety constraint function are met. In this paper, a novel BO-based approach is introduced that satisfies safety requirements irrespective of properties of the constraint function. This strong theoretical guarantee is obtained at the cost of allowing for an arbitrary, controllable but non-zero, rate of violation of the safety constraint. The proposed method, referred to as SAFE-BOCP, builds on online conformal prediction (CP) and is specialized to the cases in which feedback on the safety constraint is either noiseless or noisy. Experimental results on synthetic and real-world data validate the advantages and flexibility of the proposed SAFE-BOCP.

Bayesian Optimization with Formal Safety Guarantees via Online Conformal Prediction

TL;DR

The paper addresses constrained black-box zero-th order optimization where a safety constraint must be respected while maximizing . It introduces SAFE-BOCP, combining Bayesian optimization with online conformal prediction to produce an adaptive, pessimistic safe set that guarantees safety irrespective of the constraint function, at the cost of a non-zero target safety violation rate . Two algorithms are proposed: deterministic D-SAFE-BOCP and probabilistic P-SAFE-BOCP, each with formal safety guarantees (the latter under noise) and a data-driven tail-bound mechanism for practical resilience. The methods are validated on synthetic benchmarks and real-world tasks (e.g., movie recommendations and chemical reactor optimization), demonstrating competitive optimization performance while controlling safety risk and without strong model assumptions about . This work broadens the applicability of safe BO to safety-critical domains by providing assumption-free safety guarantees and practical probability bounds.

Abstract

Black-box zero-th order optimization is a central primitive for applications in fields as diverse as finance, physics, and engineering. In a common formulation of this problem, a designer sequentially attempts candidate solutions, receiving noisy feedback on the value of each attempt from the system. In this paper, we study scenarios in which feedback is also provided on the safety of the attempted solution, and the optimizer is constrained to limit the number of unsafe solutions that are tried throughout the optimization process. Focusing on methods based on Bayesian optimization (BO), prior art has introduced an optimization scheme -- referred to as SAFEOPT -- that is guaranteed not to select any unsafe solution with a controllable probability over feedback noise as long as strict assumptions on the safety constraint function are met. In this paper, a novel BO-based approach is introduced that satisfies safety requirements irrespective of properties of the constraint function. This strong theoretical guarantee is obtained at the cost of allowing for an arbitrary, controllable but non-zero, rate of violation of the safety constraint. The proposed method, referred to as SAFE-BOCP, builds on online conformal prediction (CP) and is specialized to the cases in which feedback on the safety constraint is either noiseless or noisy. Experimental results on synthetic and real-world data validate the advantages and flexibility of the proposed SAFE-BOCP.
Paper Structure (34 sections, 6 theorems, 50 equations, 10 figures, 1 table, 3 algorithms)

This paper contains 34 sections, 6 theorems, 50 equations, 10 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Context and Scope
  3. Related Work
  4. Main Contributions
  5. Problem Formulation
  6. Optimization Problem and Safety Constraint
  7. Sequential Surrogate-Based Safe Optimization
  8. Safety Constraints
  9. Deterministic Safety Constraint
  10. Probabilistic Safety Constraint
  11. Safe Bayesian Optimization
  12. Gaussian Process
  13. Credible Intervals and Safe Set
  14. S AFEO PT
  15. Scope and Working Assumptions
  16. ...and 19 more sections

Key Result

Theorem 1

(Safety Guarantee of S AFEO PTFelix2016safeopt) Assume that the RKHS norm of the true constraint function $q(\mathbf{x})$ is bounded by $B>0$ as in eq: norm bound B. By choosing the scaling parameter $\beta_{t+1}$ as in eq: rkhs beta, S AFEO PT satisfies the probabilistic safety constraint eq: proba

Figures (10)

  • Figure 1: This paper studies black-box zero-th order optimization with safety constraints. At each step $t=1,2,...$ of the sequential optimization process, the optimizer selects a candidate solution $\mathbf{x}_t$ and receives noisy feedback on the values of the objective function $f(\mathbf{x}_t)$ and of the constraint function $q(\mathbf{x}_t)$. Candidate solutions $\mathbf{x}_t$ yielding a negative value for the constraint function, $q(\mathbf{x}_t)<0$, are deemed to be unsafe. We wish to keep the safety violation rate, i.e., the fraction of unsafe solutions attempted during the optimization process, below some tolerated threshold.
  • Figure 2: Block diagram of Safe-BO schemes consisting of the main steps of safe set creation, producing the safe set $\mathcal{S}_{t+1}$, and of acquisition, selecting the next iterate $\mathbf{x}_{t+1}$.
  • Figure 3: Function $\beta_t=\varphi(\Delta\alpha_t)$ in \ref{['eq: q func']}, which determines the scaling factor $\beta_t$ as a function of the excess violation rate $\Delta \alpha_t$.
  • Figure 4: Violation-rate$(t)$ (top) and optimality ratio (bottom) against iteration index $t$ with target violation rate $\alpha=0.3$ (dot-dashed line), update rate $\eta=2$, misspecified kernel bandwidth $h=1/14.58$, RKHS norm bound $B=||q||_{\kappa^*}$ and total number of iteration $T=50$.
  • Figure 5: Violation rate \ref{['eq:goal']} (top) and optimality ratio \ref{['eq: optimality ratio']} (bottom) against the ratio between the RKHS norm bound $B$ assumed by GP and the actual norm $||q||_{\kappa^*}$ in \ref{['eq: norm bound B']}. The dashed lines are obtained with well-specified GP models, which corresponds to kernel bandwidth $h=1/1.69$ (same one for $\kappa^*(\mathbf{x},\mathbf{x}')$), while the solid lines are obtained with misspecified GP models, having kernel bandwidth $h=1/14.58$.
  • ...and 5 more figures

Theorems & Definitions (9)

  • Theorem 1
  • Theorem 2: Safety Guarantee of D-S AFE-B OCP
  • proof
  • Lemma 1: Estimated Violation Rate
  • proof
  • Theorem 3: Safety Guarantee of P-S AFE-B OCP
  • proof
  • Lemma 2: Estimated Upper Bound
  • Corollary 1