Bayesian Optimization of Function Networks

TL;DR

This paper introduces a Bayesian optimization method for networks of functions that efficiently uses intermediate outputs, improving query efficiency and convergence in complex, time-consuming evaluation scenarios.

Contribution

It develops a novel Bayesian optimization approach leveraging intermediate network outputs and Gaussian process modeling, with proven asymptotic optimality and practical performance gains.

Findings

Outperforms standard Bayesian optimization in synthetic problems

Achieves asymptotic convergence to the global optimum

Utilizes intermediate outputs for improved efficiency

Abstract

We consider Bayesian optimization of the output of a network of functions, where each function takes as input the output of its parent nodes, and where the network takes significant time to evaluate. Such problems arise, for example, in reinforcement learning, engineering design, and manufacturing. While the standard Bayesian optimization approach observes only the final output, our approach delivers greater query efficiency by leveraging information that the former ignores: intermediate output within the network. This is achieved by modeling the nodes of the network using Gaussian processes and choosing the points to evaluate using, as our acquisition function, the expected improvement computed with respect to the implied posterior on the objective. Although the non-Gaussian nature of this posterior prevents computing our acquisition function in closed form, we show that it can be efficiently maximized via sample average approximation. In addition, we prove that our method is asymptotically consistent, meaning that it finds a globally optimal solution as the number of evaluations grows to infinity, thus generalizing previously known convergence results for the expected improvement. Notably, this holds even though our method might not evaluate the domain densely, instead leveraging problem structure to leave regions unexplored. Finally, we show that our approach dramatically outperforms standard Bayesian optimization methods in several synthetic and real-world problems.