[Paper Review] Bayesian Optimization with Output-Weighted Importance Sampling.

Antoine Blanchard, Themistoklis P. Sapsis|arXiv (Cornell University)|Apr 22, 2020

Gaussian Processes and Bayesian Inference16 references4 citations

TL;DR

This paper proposes a Bayesian optimization framework that uses likelihood ratios as output-weighted importance sampling to guide search toward inputs yielding abnormally large or small objective values. By weighting acquisition functions via likelihood ratios, the method improves optimization efficiency and performance over unweighted approaches, especially in high-dimensional settings.

ABSTRACT

In Bayesian optimization, accounting for the importance of the output relative to the input is a crucial yet challenging exercise, as it can considerably improve the final result but often involves inaccurate and cumbersome entropy estimations. We approach the problem from the perspective of importance-sampling theory, and advocate the use of the likelihood ratio to guide the search algorithm towards regions of the input space where the objective function assumes abnormally large or small values. The likelihood ratio acts as a sampling weight and can be computed at each iteration without severely deteriorating the overall efficiency of the algorithm. In particular, it can be approximated in a way that makes the approach tractable in high dimensions. The likelihood-weighted acquisition functions introduced in this work are found to outperform their unweighted counterparts in a number of applications.

Motivation & Objective

• To address the challenge of incorporating output importance into Bayesian optimization without relying on inaccurate entropy estimations.
• To develop a tractable method for weighting search based on the likelihood ratio of objective function values.
• To improve optimization performance by focusing on regions where outputs deviate significantly from the norm.
• To enable efficient application in high-dimensional input spaces through approximation of likelihood weights.

Proposed method

• Uses the likelihood ratio as a sampling weight to emphasize inputs that produce extreme objective values.
• Derives acquisition functions weighted by the likelihood ratio to guide the optimization process.
• Approximates the likelihood ratio at each iteration to maintain computational efficiency.
• Applies importance-sampling theory to reweight the search distribution based on output significance.
• Integrates the weighted acquisition function into a standard Bayesian optimization loop with Gaussian process priors.
• Enables scalability to high dimensions through tractable approximation of the likelihood ratio.

Experimental results

Research questions

• RQ1 Can likelihood ratios be used effectively to weight Bayesian optimization acquisitions based on output importance?
• RQ2 How does output-weighted sampling compare to standard acquisition functions in terms of convergence and performance?
• RQ3 Can the likelihood-weighted approach maintain efficiency while improving search for extreme objective values?
• RQ4 What is the impact of likelihood-weighted sampling in high-dimensional optimization problems?
• RQ5 How accurately can the likelihood ratio be approximated without degrading optimization performance?

Key findings

• The likelihood-weighted acquisition functions outperform their unweighted counterparts across multiple benchmark applications.
• The method effectively guides the search toward inputs that yield abnormally large or small objective values.
• The likelihood ratio can be computed and approximated efficiently at each iteration without significant computational overhead.
• The approach remains tractable and effective in high-dimensional input spaces.
• The use of output-weighted importance sampling improves optimization efficiency by focusing on functionally significant regions of the input space.
• Empirical results demonstrate consistent performance gains over standard Bayesian optimization under various settings.