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Expected Diverse Utility (EDU): Diverse Bayesian Optimization of Expensive Computer Simulators

John Joshua Miller, Simon Mak, Benny Sun, Sai Ranjeet Narayanan, Suo Yang, Zongxuan Sun, Kenneth S. Kim, Chol-Bum Mike Kweon

TL;DR

A new Expected Diverse Utility (EDU) method that searches for diverse ``$epsilon$-optimal'' solutions: locally-optimal solutions within a tolerance level $\epsilon>0$ from a global optimum, and reveals a novel exploration-exploitation-diversity trade-off, which incorporates the desired diversity property within the well-known exploration-exploitation trade-off.

Abstract

The optimization of expensive black-box simulators arises in a myriad of modern scientific and engineering applications. Bayesian optimization provides an appealing solution, by leveraging a fitted surrogate model to guide the selection of subsequent simulator evaluations. In practice, however, the objective is often not to obtain a single good solution, but rather a ``basket'' of good solutions from which users can choose for downstream decision-making. This need arises in our motivating application for real-time control of internal combustion engines for flight propulsion, where a diverse set of control strategies is essential for stable flight control. There has been little work on this front for Bayesian optimization. We thus propose a new Expected Diverse Utility (EDU) method that searches for diverse ``$ε$-optimal'' solutions: locally-optimal solutions within a tolerance level $ε> 0$ from a global optimum. We show that EDU yields a closed-form acquisition function under a Gaussian process surrogate model, which facilitates efficient sequential queries via automatic differentiation. This closed form further reveals a novel exploration-exploitation-diversity trade-off, which incorporates the desired diversity property within the well-known exploration-exploitation trade-off. We demonstrate the improvement of EDU over existing methods in a suite of numerical experiments, then explore the EDU in two applications on rover trajectory optimization and engine control for flight propulsion.

Expected Diverse Utility (EDU): Diverse Bayesian Optimization of Expensive Computer Simulators

TL;DR

A new Expected Diverse Utility (EDU) method that searches for diverse ``-optimal'' solutions: locally-optimal solutions within a tolerance level from a global optimum, and reveals a novel exploration-exploitation-diversity trade-off, which incorporates the desired diversity property within the well-known exploration-exploitation trade-off.

Abstract

The optimization of expensive black-box simulators arises in a myriad of modern scientific and engineering applications. Bayesian optimization provides an appealing solution, by leveraging a fitted surrogate model to guide the selection of subsequent simulator evaluations. In practice, however, the objective is often not to obtain a single good solution, but rather a ``basket'' of good solutions from which users can choose for downstream decision-making. This need arises in our motivating application for real-time control of internal combustion engines for flight propulsion, where a diverse set of control strategies is essential for stable flight control. There has been little work on this front for Bayesian optimization. We thus propose a new Expected Diverse Utility (EDU) method that searches for diverse ``-optimal'' solutions: locally-optimal solutions within a tolerance level from a global optimum. We show that EDU yields a closed-form acquisition function under a Gaussian process surrogate model, which facilitates efficient sequential queries via automatic differentiation. This closed form further reveals a novel exploration-exploitation-diversity trade-off, which incorporates the desired diversity property within the well-known exploration-exploitation trade-off. We demonstrate the improvement of EDU over existing methods in a suite of numerical experiments, then explore the EDU in two applications on rover trajectory optimization and engine control for flight propulsion.
Paper Structure (21 sections, 1 theorem, 16 equations, 12 figures, 3 tables, 1 algorithm)

This paper contains 21 sections, 1 theorem, 16 equations, 12 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Real-Time Engine Control for Sustainable Aviation
  3. Problem Background
  4. A Need for Diverse Solutions
  5. Expected Diverse Utility
  6. Gaussian Process Modeling
  7. Diverse Utility
  8. Acquisition Function
  9. Methodological Developments
  10. Batch EDU
  11. Acquisition Optimization
  12. Tolerance Specification and Determination
  13. Algorithm Statement
  14. Numerical Experiments
  15. $2^d$-Bowls Function
  16. ...and 6 more sections

Key Result

Proposition 1

Conditional on data $\mathcal{D}_n$, the EDU acquisition function takes the form: where $\zeta_n = (\gamma_n-\mu_n(\mathbf{x}))/\sigma_n(\mathbf{x})$.

Figures (12)

  • Figure 1: Schematic of the considered metal engine for CFD simulation. Here, the piston bowl and cylinder head are shown, and fuel injection occurs via seven nozzles (marked by white cones). The glow plug (in blue) is seen attached to the cylinder head.
  • Figure 2: Visualizing a function with $K=2$ diverse optima. Here, with $\epsilon = 0.15$, there are two resulting subregions $\mathcal{R}_{\epsilon,1}$ and $\mathcal{R}_{\epsilon,2}$, with its corresponding minimizers $\mathbf{x}^*_1$ and $\mathbf{x}^*_2$.
  • Figure 3: Visualizing the 2-d four-bowls experiment for diverse optimization. Black points show the $n=10$ initial design points that are shared by all methods. Blue points show the sequential points selected by each method. Red points mark the samples that are within the $\epsilon$-optimal subregion $\mathcal{R}_{\epsilon,k}$ for one of the four optimal solutions.
  • Figure 4: Visualizing the intuition behind the DU utility function in \ref{['eq:di']}.
  • Figure 5: Comparing the EI and EDU acquisition functions on the four-bowls function (see Section \ref{['sec:bowls']}) with $n=15$ initial design points. (Left) Initial design points in black, overlaid on function contours of $f$. (Middle and Right) Contours of the EI and EDU acquisitions, respectively, with its corresponding maximizers.
  • ...and 7 more figures

Theorems & Definitions (2)

  • Proposition 1
  • proof