Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Randomized Kiring Believer for Parallel Bayesian Optimization with Regret Bounds

Shuhei Sugiura, Ichiro Takeuchi, Shion Takeno

TL;DR

This study proposes a PBO method, called randomized kriging believer (KB), based on a well-known KB heuristic and inheriting the advantages of the original KB: low computational complexity, a simple implementation, versatility across various BO methods, and applicability to asynchronous parallelization.

Abstract

We consider an optimization problem of an expensive-to-evaluate black-box function, in which we can obtain noisy function values in parallel. For this problem, parallel Bayesian optimization (PBO) is a promising approach, which aims to optimize with fewer function evaluations by selecting a diverse input set for parallel evaluation. However, existing PBO methods suffer from poor practical performance or lack theoretical guarantees. In this study, we propose a PBO method, called randomized kriging believer (KB), based on a well-known KB heuristic and inheriting the advantages of the original KB: low computational complexity, a simple implementation, versatility across various BO methods, and applicability to asynchronous parallelization. Furthermore, we show that our randomized KB achieves Bayesian expected regret guarantees. We demonstrate the effectiveness of the proposed method through experiments on synthetic and benchmark functions and emulators of real-world data.

Randomized Kiring Believer for Parallel Bayesian Optimization with Regret Bounds

TL;DR

This study proposes a PBO method, called randomized kriging believer (KB), based on a well-known KB heuristic and inheriting the advantages of the original KB: low computational complexity, a simple implementation, versatility across various BO methods, and applicability to asynchronous parallelization.

Abstract

We consider an optimization problem of an expensive-to-evaluate black-box function, in which we can obtain noisy function values in parallel. For this problem, parallel Bayesian optimization (PBO) is a promising approach, which aims to optimize with fewer function evaluations by selecting a diverse input set for parallel evaluation. However, existing PBO methods suffer from poor practical performance or lack theoretical guarantees. In this study, we propose a PBO method, called randomized kriging believer (KB), based on a well-known KB heuristic and inheriting the advantages of the original KB: low computational complexity, a simple implementation, versatility across various BO methods, and applicability to asynchronous parallelization. Furthermore, we show that our randomized KB achieves Bayesian expected regret guarantees. We demonstrate the effectiveness of the proposed method through experiments on synthetic and benchmark functions and emulators of real-world data.
Paper Structure (34 sections, 19 theorems, 101 equations, 4 figures, 1 algorithm)

This paper contains 34 sections, 19 theorems, 101 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Joint Selection Methods.
  4. Greedy Selection Methods.
  5. Distributed Selection Methods.
  6. Regret Analysis for PBO.
  7. Preliminaries
  8. Problem Statement
  9. Gaussian Process Regression
  10. Acquisition Functions for BO
  11. Kriging Believer for PBO
  12. Bayesian Regret and Maximum Information Gain
  13. Randomized Kriging Believer
  14. Regret Analysis
  15. Theoretical Conditions for Base BO Algorithms
  16. ...and 19 more sections

Key Result

Lemma 4.1

Suppose that Assumption asm_basic and Condition cnd_alg hold. Let $\bm x_t=\mathcal{A}{\left(\mathcal{D}_{t-1}\right)}$. Then, the following holds: where $B_T$ is defined using $C_1=2/\log{\left(1+\sigma_{\rm noise}^{-2}\right)}$ as

Figures (4)

  • Figure 1: Schematic illustration of the proposed method for three consecutive iterations. We consider the parallel optimization problem, in which multiple evaluations can be performed simultaneously. For efficient optimization, it is necessary to evaluate a diverse input set while avoiding redundant evaluation. The left figure shows iteration 1, where no inputs are under evaluation, and thus, the next input is chosen as in standard BO. The middle and right figures show iterations 2 and 3, respectively. There are inputs currently being evaluated, indicated by dashed lines. In these cases, RKB generates fantasized data from the predictive distribution at the inputs under evaluation. Then, the next input is selected by maximizing an AF based on the model trained on both the observed (real) data and the fantasized data. As a result, BO avoids redundant evaluation and ensures diversity among the evaluated inputs.
  • Figure 2: Average and standard error of simple regret across synthetic function experiments. One batch corresponds to 8 iterations.
  • Figure 3: Average and standard error of simple regret across benchmark function experiments. One batch corresponds to 8 iterations.
  • Figure 4: Average and standard error of objective value across emulator experiments. One batch corresponds to 8 iterations.

Theorems & Definitions (31)

  • Definition 2.1
  • Lemma 4.1: BCR bound for sequential optimization
  • Theorem 4.1: BCR bound for finite input domains
  • Theorem 4.2: BCR bound for continuous input domains
  • Theorem 4.3: BSR bound
  • Lemma A.1
  • proof : Proof of Lemma \ref{['variance_bound']}
  • Lemma A.1: BCR bound for sequential optimization
  • proof : Proof of Lemma \ref{['general_RA']}
  • Lemma B.1
  • ...and 21 more