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Pareto Front-Diverse Batch Multi-Objective Bayesian Optimization

Alaleh Ahmadianshalchi, Syrine Belakaria, Janardhan Rao Doppa

TL;DR

PDBO employs a multi-armed bandit approach to select one acquisition function from a given library and utilizes Determinantal Point Processes to choose a Pareto-front-diverse batch of inputs for evaluation from the candidate set obtained from the first step.

Abstract

We consider the problem of multi-objective optimization (MOO) of expensive black-box functions with the goal of discovering high-quality and diverse Pareto fronts where we are allowed to evaluate a batch of inputs. This problem arises in many real-world applications including penicillin production where diversity of solutions is critical. We solve this problem in the framework of Bayesian optimization (BO) and propose a novel approach referred to as Pareto front-Diverse Batch Multi-Objective BO (PDBO). PDBO tackles two important challenges: 1) How to automatically select the best acquisition function in each BO iteration, and 2) How to select a diverse batch of inputs by considering multiple objectives. We propose principled solutions to address these two challenges. First, PDBO employs a multi-armed bandit approach to select one acquisition function from a given library. We solve a cheap MOO problem by assigning the selected acquisition function for each expensive objective function to obtain a candidate set of inputs for evaluation. Second, it utilizes Determinantal Point Processes (DPPs) to choose a Pareto-front-diverse batch of inputs for evaluation from the candidate set obtained from the first step. The key parameters for the methods behind these two steps are updated after each round of function evaluations. Experiments on multiple MOO benchmarks demonstrate that PDBO outperforms prior methods in terms of both the quality and diversity of Pareto solutions.

Pareto Front-Diverse Batch Multi-Objective Bayesian Optimization

TL;DR

PDBO employs a multi-armed bandit approach to select one acquisition function from a given library and utilizes Determinantal Point Processes to choose a Pareto-front-diverse batch of inputs for evaluation from the candidate set obtained from the first step.

Abstract

We consider the problem of multi-objective optimization (MOO) of expensive black-box functions with the goal of discovering high-quality and diverse Pareto fronts where we are allowed to evaluate a batch of inputs. This problem arises in many real-world applications including penicillin production where diversity of solutions is critical. We solve this problem in the framework of Bayesian optimization (BO) and propose a novel approach referred to as Pareto front-Diverse Batch Multi-Objective BO (PDBO). PDBO tackles two important challenges: 1) How to automatically select the best acquisition function in each BO iteration, and 2) How to select a diverse batch of inputs by considering multiple objectives. We propose principled solutions to address these two challenges. First, PDBO employs a multi-armed bandit approach to select one acquisition function from a given library. We solve a cheap MOO problem by assigning the selected acquisition function for each expensive objective function to obtain a candidate set of inputs for evaluation. Second, it utilizes Determinantal Point Processes (DPPs) to choose a Pareto-front-diverse batch of inputs for evaluation from the candidate set obtained from the first step. The key parameters for the methods behind these two steps are updated after each round of function evaluations. Experiments on multiple MOO benchmarks demonstrate that PDBO outperforms prior methods in terms of both the quality and diversity of Pareto solutions.
Paper Structure (24 sections, 5 theorems, 28 equations, 15 figures, 4 tables, 3 algorithms)

This paper contains 24 sections, 5 theorems, 28 equations, 15 figures, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Problem Setup and Background
  3. Bayesian Optimization Framework
  4. Related Work
  5. Proposed PDBO Algorithm
  6. Multi-arm Bandit Strategy for Adaptive Acquisition Function Selection
  7. DPPs for Batch Selection
  8. Theoretical Analysis
  9. Analysis and Ablation Study
  10. Experiments and Results
  11. Summary
  12. Appendix
  13. Theoretical Analysis
  14. PDBO Hypervolume Experiments
  15. PDBO Diversity Experiments
  16. ...and 9 more sections

Key Result

Theorem 5.1

Let $\mathbf{x}^*$ be a point in the optimal Pareto set $\mathcal{X^*}$. Let $\mathbf{x}$ be a point in the Pareto set $\mathcal{X}_t$ estimated by PDBO via solving cheap MOO problem at the $t^{th}$ iteration. Let the cumulative regret for the multiple objectives be defined as $R_{T_{max}}(\mathbf{x

Figures (15)

  • Figure 1: High-level overview of PDBO algorithm illustrating its three key components.
  • Figure 2: Diverse Pareto front (DPF) results evaluated on multiple benchmarks and batch sizes.
  • Figure 3: Hypervolume results evaluated on multiple benchmarks and batch sizes.
  • Figure 4: The Pareto front of the inputs selected by each algorithm. Points from later batches are depicted in darker colors.
  • Figure 5: Comparing PDBO against PDBO with the sum of posterior means (SPM) as the reward.
  • ...and 10 more figures

Theorems & Definitions (6)

  • Theorem 5.1
  • Lemma 8.1
  • Lemma 8.2
  • Lemma 8.3
  • Lemma 8.4
  • proof : Proof of Theorem \ref{['theorem']}