[Paper Review] Noisy-Input Entropy Search for Efficient Robust Bayesian Optimization

Lukas P. Fröhlich, Edgar D. Klenske+3 more|arXiv (Cornell University)|Feb 7, 2020

Gaussian Processes and Bayesian Inference34 references18 citations

TL;DR

This paper introduces Noisy-Input Entropy Search (NES), an information-theoretic acquisition function for robust Bayesian optimization that accounts for input uncertainty by modeling the robust objective as a Gaussian process. NES maximizes information gain about the robust optimum and outperforms existing methods on benchmark and aerospace tasks, achieving significantly lower inference regret by reliably finding broad, stable optima despite input noise.

ABSTRACT

We consider the problem of robust optimization within the well-established Bayesian optimization (BO) framework. While BO is intrinsically robust to noisy evaluations of the objective function, standard approaches do not consider the case of uncertainty about the input parameters. In this paper, we propose Noisy-Input Entropy Search (NES), a novel information-theoretic acquisition function that is designed to find robust optima for problems with both input and measurement noise. NES is based on the key insight that the robust objective in many cases can be modeled as a Gaussian process, however, it cannot be observed directly. We evaluate NES on several benchmark problems from the optimization literature and from engineering. The results show that NES reliably finds robust optima, outperforming existing methods from the literature on all benchmarks.

Motivation & Objective

• To address the limitation of standard Bayesian optimization in handling input uncertainty, where perturbations at deployment can degrade performance.
• To develop an information-theoretic acquisition function that explicitly maximizes knowledge about the robust optimum under input noise.
• To model the robust objective as a latent Gaussian process that cannot be directly observed but must be inferred from noisy function evaluations.
• To enable efficient and reliable optimization in real-world applications where input parameters are subject to random perturbations at deployment.
• To outperform heuristic and non-information-theoretic robust BO methods through a principled, sample-efficient approach.

Proposed method

• Proposes Noisy-Input Entropy Search (NES), an acquisition function that maximizes information gain about the robust maximum value $ g^* $, defined as the expected objective value under input perturbations.
• Models the robust objective as a Gaussian process (GP), even though it is not directly observable, and conditions the GP predictive distribution on the robust optimum.
• Uses max-value entropy search (MES) as a foundation to enable efficient computation of information gain.
• Develops two approximation methods for the intractable predictive distribution conditioned on $ g^* $: rejection sampling (exact in the limit) and expectation propagation (computationally efficient).
• Infers kernel hyperparameters and observation noise via marginal likelihood maximization after each evaluation, with a log-normal hyperprior on lengthscales to stabilize convergence.
• Employs a probabilistic formulation of robustness where optima are preferred if they remain high under random input perturbations, favoring broad over narrow peaks.

Experimental results

Research questions

• RQ1 Can an information-theoretic acquisition function be designed to robustly optimize black-box functions when inputs are subject to random perturbations at deployment?
• RQ2 How can the predictive distribution of a GP be conditioned on the robust maximum value when this value is not directly observable?
• RQ3 Does an entropy-based acquisition function that maximizes information about the robust optimum outperform heuristic or non-information-theoretic robust BO methods?
• RQ4 How does the proposed method scale in performance with increasing input dimensionality and noise levels?
• RQ5 Can the method reliably find robust optima in real-world engineering applications with high-stakes consequences, such as aerospace trajectory planning?

Key findings

• NES consistently outperforms standard Bayesian optimization and existing robust methods on all benchmark functions, achieving superior convergence and lower inference regret.
• On the 3-dimensional Hartmann function with $ m{ ext{Σ}}_x = 0.1^2 \mathbf{I} $, NES achieves significantly lower inference regret than standard BO and other robust methods.
• In the gravity assist maneuver task, NES-EP reduces inference regret by nearly two orders of magnitude compared to standard BO after 15 evaluations.
• The performance of NES is robust across different benchmark types, including 1D, 2D, and 3D functions with varying noise levels and objective shapes.
• The expectation propagation approximation (NES-EP) provides a computationally efficient alternative to rejection sampling while maintaining high accuracy and performance.
• Standard BO, while finding the global optimum, fails in the robust setting due to sensitivity to input perturbations, as evidenced by stagnating inference regret in the gravity assist task.