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Multi-Fidelity Bayesian Optimization With Across-Task Transferable Max-Value Entropy Search

Yunchuan Zhang, Sangwoo Park, Osvaldo Simeone

TL;DR

The paper tackles sequential, costly black-box optimization under multi-fidelity cost structures across related tasks. It extends max-value entropy search by introducing shared GP parameters $\boldsymbol{\theta}$ that encapsulate transferable knowledge, and it uses particle-based variational updates (SVGD) to propagate this information across tasks. The proposed Multi-Fidelity Transferable MES (MFT-MES) augments the current-task information gain with a term that captures learning about $\boldsymbol{\theta}$, enabling faster convergence on future tasks. Theoretical regret bounds show how transferring knowledge reduces cumulative regret as the number of tasks grows, and empirical results on synthetic benchmarks and radio-resource management demonstrate substantial gains over single-task MF-MES and non-transfer baselines as more tasks are processed.

Abstract

In many applications, ranging from logistics to engineering, a designer is faced with a sequence of optimization tasks for which the objectives are in the form of black-box functions that are costly to evaluate. Furthermore, higher-fidelity evaluations of the optimization objectives often entail a larger cost. Existing multi-fidelity black-box optimization strategies select candidate solutions and fidelity levels with the goal of maximizing the information about the optimal value or the optimal solution for the current task. Assuming that successive optimization tasks are related, this paper introduces a novel information-theoretic acquisition function that balances the need to acquire information about the current task with the goal of collecting information transferable to future tasks. The proposed method transfers across tasks distributions over parameters of a Gaussian process surrogate model by implementing particle-based variational Bayesian updates. Theoretical insights based on the analysis of the expected regret substantiate the benefits of acquiring transferable knowledge across tasks. Furthermore, experimental results across synthetic and real-world examples reveal that the proposed acquisition strategy that caters to future tasks can significantly improve the optimization efficiency as soon as a sufficient number of tasks is processed.

Multi-Fidelity Bayesian Optimization With Across-Task Transferable Max-Value Entropy Search

TL;DR

The paper tackles sequential, costly black-box optimization under multi-fidelity cost structures across related tasks. It extends max-value entropy search by introducing shared GP parameters that encapsulate transferable knowledge, and it uses particle-based variational updates (SVGD) to propagate this information across tasks. The proposed Multi-Fidelity Transferable MES (MFT-MES) augments the current-task information gain with a term that captures learning about , enabling faster convergence on future tasks. Theoretical regret bounds show how transferring knowledge reduces cumulative regret as the number of tasks grows, and empirical results on synthetic benchmarks and radio-resource management demonstrate substantial gains over single-task MF-MES and non-transfer baselines as more tasks are processed.

Abstract

In many applications, ranging from logistics to engineering, a designer is faced with a sequence of optimization tasks for which the objectives are in the form of black-box functions that are costly to evaluate. Furthermore, higher-fidelity evaluations of the optimization objectives often entail a larger cost. Existing multi-fidelity black-box optimization strategies select candidate solutions and fidelity levels with the goal of maximizing the information about the optimal value or the optimal solution for the current task. Assuming that successive optimization tasks are related, this paper introduces a novel information-theoretic acquisition function that balances the need to acquire information about the current task with the goal of collecting information transferable to future tasks. The proposed method transfers across tasks distributions over parameters of a Gaussian process surrogate model by implementing particle-based variational Bayesian updates. Theoretical insights based on the analysis of the expected regret substantiate the benefits of acquiring transferable knowledge across tasks. Furthermore, experimental results across synthetic and real-world examples reveal that the proposed acquisition strategy that caters to future tasks can significantly improve the optimization efficiency as soon as a sufficient number of tasks is processed.
Paper Structure (27 sections, 1 theorem, 59 equations, 7 figures, 2 algorithms)

This paper contains 27 sections, 1 theorem, 59 equations, 7 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Context and Scope
  3. Related Work
  4. Main Contributions
  5. Organization
  6. Problem Definition And Preliminaries
  7. Sequential Multi-Task Black-Box Optimization
  8. Gaussian Process
  9. Multi-fidelity Gaussian Process
  10. Single-Task Multi-fidelity Bayesian Optimization
  11. Multi-Fidelity Max-Value Entropy Search
  12. Optimizing the Parameter Vector $\boldsymbol{\theta}$
  13. Sequential Multi-fidelity Bayesian Optimization With Transferable Max-Value Entropy Search
  14. Multi-Fidelity Transferable Max-Value Entropy Search
  15. Bayesian Learning for the Parameter Vector $\boldsymbol{\theta}$
  16. ...and 12 more sections

Key Result

Theorem 1

Under Assumptions assump:regret_cond and lemma_3, denoting $\varepsilon=\varepsilon_1+\varepsilon_2$, the cumulative expected regret of the two-phase MFT-MES scheme satisfies the inequality where $\nu_{n,t}=\min_\mathbf{x}\tilde{r}_{n,t}(\mathbf{x}|\boldsymbol{\theta}^*)$ is the minimum normalized expected regret eq: normalized regret at time $t$; $t_n^* = \arg\max_{t \in \{1,...,c\cdot T\}} \nu_

Figures (7)

  • Figure 1: This paper studies a sequential multi-task optimization setting with multi-fidelity approximations of expensive-to-evaluate black-box objective functions. For any current task $n$, over time index $t=1,2,...,T_n$, the optimizer selects a pair of query point $\mathbf{x}_{n,t}$ and fidelity level $m_{n,t}$, requiring an approximation cost $\lambda^{(m)}$. As a result, the optimizer receives noisy feedback $y_{n,t}^{(m)}$ about target objective value $f_n(\mathbf{x}_{n,t})$. We wish to approach the global optimal solution $\mathbf{x}_n^*$ of objective $f_n(\mathbf{x})$ while abiding by a total simulation cost budget $\Lambda$.
  • Figure 2: Comparison between selected prior works and the proposed MFT-MES method.
  • Figure 3: Synthetic optimization tasks: Simple regret \ref{['eq: simple regret']} against the number of tasks, $n$, for MF-MES, Continual MF-MES $(\beta=0)$ with $V=5$ and $V=10$ particles, and MFT-MES ($\beta=1.2$) with $V=5$ and $V=10$ particles.
  • Figure 4: Synthetic optimization tasks: Log-simple regret against weight parameter $\beta$ and the number of tasks, $n$, for MF-MES (black dash-dotted line), Continual MF-MES (red solid line), and MFT-MES (blue surface). The optimal values of $\beta$ at the corresponding number of tasks $n$ are labeled as gold stars. The number of particles for Continual MF-MES and MFT-MES is set to $V=10$.
  • Figure 5: Radio resource management for wireless systems zhang2023bayesian: Optimality ratio \ref{['eq: optimality ratio']} against the number of tasks, $n$, for MF-MES, Continual MF-MES ($\beta=0$), and MFT-MES ($\beta=1.6$) with $V=10$ particles.
  • ...and 2 more figures

Theorems & Definitions (3)

  • Definition 1: $\varepsilon$-sharpness
  • Definition 2: $\varepsilon$-stability
  • Theorem 1: Cumulative Expected Regret Bound of MFT-MES