A Generalized Framework of Multifidelity Max-Value Entropy Search Through Joint Entropy

Abstract

Bayesian optimization (BO) is a popular method for expensive black-box optimization problems; however, querying the objective function at every iteration can be a bottleneck that hinders efficient search capabilities. In this regard, multifidelity Bayesian optimization (MFBO) aims to accelerate BO by incorporating lower-fidelity observations available with a lower sampling cost. In our previous work, we proposed an information-theoretic approach to MFBO, referred to as multifidelity max-value entropy search (MF-MES), which inherits practical effectiveness and computational simplicity of the well-known max-value entropy search (MES) for the single-fidelity BO. However, the applicability of MF-MES is still limited to the case that a single observation is sequentially obtained. In this letter, we generalize MF-MES so that information gain can be evaluated even when multiple observations are simultaneously obtained. This generalization enables MF-MES to address two practical problem settings: synchronous parallelization and trace-aware querying. We show that the acquisition functions for these extensions inherit the simplicity of MF-MES without introducing additional assumptions. We also provide computational techniques for entropy evaluation and posterior sampling in the acquisition functions, which can be commonly used for all variants of MF-MES. The effectiveness of MF-MES is demonstrated using benchmark functions and real-world applications such as materials science data and hyperparameter tuning of machine-learning algorithms.

GFlowNets for AI-driven scientific discovery

“…Additionally, we might have access to oracles with different costs and fidelities to evaluate f; for example, to obtain the simulated binding affinity of a molecule to a protein, oracles can include free energy perturbation and molecular docking, where the former is substantially more accurate but the computational cost is orders of magnitude higher.. This setting is studied in Multi-fidelity Bayesian optimization (Picheny et al, 2010;Kandasamy et al, 2017;Takeno et al, 2020). Another important aspect in practical applications is multiple objectives.…”

Active Learning and Bayesian Optimization: a Unified Perspective to Learn with a Goal

“…This is obtained through an adaptive sampling scheme driven by the multifidelity acquisition function U(x, l) that extends the infill criteria of Bayesian optimization, and selects the pair of sample and the associated level of fidelity to query (x new , l new ) ∈ max x∈χ,l∈L U(x, l) that is likely to provide higher gains with a regard for the computational expenditure. Among different formulations, well known multifidelity acquisition functions to address optimization problems are the Multifidelity Probability of Improvement (MFPI) [62], Multifidelity Expected Improvement (MFEI) [63], Multifidelity Predictive Entropy Search (MFPES) [64], Multifidelity Max-Value Entropy Search (MFMES) [65], and non-myopic multifidelity expected improvement [66]. These formulations of the acquisition function define adaptive sampling schemes that retain the infill principles characterizing the single-fidelity counterpart, and account for the dual decision task balancing the accuracy achieved through accurate queries with the associated cost during the optimization procedure.…”

“…The Multifidelity Max-Value Entropy Search (MFMES) acquisition function can be formulated extending the maxvalue entropy search to a multifidelity setting as follows [65]:…”

“…where the differential entropy is measured on the minimum value of the objective function f * L considering the highfidelity representation L. In this case, the approximation of the expectation term in Equation ( 19) relies on a Monte Carlo strategy that allows to contain the computational cost if compared with the procedure used for the MFES acquisition function [65].…”

Investigating Bayesian optimization for expensive-to-evaluate black box functions: Application in fluid dynamics

“…Information-based policies, in particular those using entropy, aim to find a candidate point x that reduces the entropy of the posterior distribution p(x|D n ). While Entropy Search (ES) and Predictive Entropy Search (PES) are computationally expensive, Wang and Jegelka [14] and Takeno et al [15] introduce Max-value Entropy Search (MES), a method that uses information about simple to compute maximal response values instead of costly to compute entropies.…”