Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Traveling Bandit: A Framework for Bayesian Optimization with Movement Costs

Qiyuan Chen, Raed Al Kontar

TL;DR

This work addresses Bayesian Optimization in settings where changing inputs incurs movement costs, by introducing a plug-in framework that first selects a batch of candidate designs via a standard BO acquisition function and then visits them along a shortest-path tour computed by a Traveling Salesman Problem. The key idea is to decouple function optimization from movement planning, enabling sublinear growth of movement costs $C_T$ while preserving sublinear regret $R_T$, with overall loss $L_T=R_T+C_T$. The authors establish a general path-length bound on tours in metric spaces using Minkowski dimension, and show that batched algorithms with $N(T)=o(T^{1/d})$ batches achieve convergence in both movement costs and regret, with concrete results for BO and other stochastic bandits (MAB and Lipschitz bandits). Empirically, the method reduces movement costs over time across multiple test functions without sacrificing regret, and is shown to be compatible with various acquisition functions and batched strategies. This framework has broad applicability to online decision-making problems where input changes carry non-negligible costs.

Abstract

This paper introduces a framework for Bayesian Optimization (BO) with metric movement costs, addressing a critical challenge in practical applications where input alterations incur varying costs. Our approach is a convenient plug-in that seamlessly integrates with the existing literature on batched algorithms, where designs within batches are observed following the solution of a Traveling Salesman Problem. The proposed method provides a theoretical guarantee of convergence in terms of movement costs for BO. Empirically, our method effectively reduces average movement costs over time while maintaining comparable regret performance to conventional BO methods. This framework also shows promise for broader applications in various bandit settings with movement costs.

The Traveling Bandit: A Framework for Bayesian Optimization with Movement Costs

TL;DR

This work addresses Bayesian Optimization in settings where changing inputs incurs movement costs, by introducing a plug-in framework that first selects a batch of candidate designs via a standard BO acquisition function and then visits them along a shortest-path tour computed by a Traveling Salesman Problem. The key idea is to decouple function optimization from movement planning, enabling sublinear growth of movement costs while preserving sublinear regret , with overall loss . The authors establish a general path-length bound on tours in metric spaces using Minkowski dimension, and show that batched algorithms with batches achieve convergence in both movement costs and regret, with concrete results for BO and other stochastic bandits (MAB and Lipschitz bandits). Empirically, the method reduces movement costs over time across multiple test functions without sacrificing regret, and is shown to be compatible with various acquisition functions and batched strategies. This framework has broad applicability to online decision-making problems where input changes carry non-negligible costs.

Abstract

This paper introduces a framework for Bayesian Optimization (BO) with metric movement costs, addressing a critical challenge in practical applications where input alterations incur varying costs. Our approach is a convenient plug-in that seamlessly integrates with the existing literature on batched algorithms, where designs within batches are observed following the solution of a Traveling Salesman Problem. The proposed method provides a theoretical guarantee of convergence in terms of movement costs for BO. Empirically, our method effectively reduces average movement costs over time while maintaining comparable regret performance to conventional BO methods. This framework also shows promise for broader applications in various bandit settings with movement costs.

Paper Structure

This paper contains 23 sections, 4 theorems, 9 equations, 5 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. BO with Movement Costs
  4. Batched Algorithms
  5. Traveling Salesman Problem
  6. Setting the Stage
  7. Problem Setup
  8. BO with a Gaussian Process Surrogate
  9. Convergence of Algorithms
  10. Batched Algorithms
  11. Asymptotic Notations
  12. A Simple Framework: Plan More and Move Less
  13. Plan-ahead with Batched Algorithms
  14. Practical Batched BO with Successive Elimination
  15. Theoretical Results: The More You Plan, the Less You Travel
  16. ...and 8 more sections

Key Result

Theorem 5

Fix any metric space $(M,c)$ of dimension $d$ satisfying Assumption asm:physical. For any set of $n$ points in $M$, we can find a path through all of them whose length is at most $O(n^{1-\frac{1}{d}}).$ This bound is tight in the sense that for any such metric space, there always exists $n$ points s

Figures (5)

  • Figure 1: Sample path of movement-cost unaware and aware approaches
  • Figure 2: Sample path of a movement cost-unaware vs. our cost-aware method.
  • Figure 3: The Failure of Movement Penalization
  • Figure 4: Regret (left) and Movement costs (right) on Synthetic Test Benchmarks
  • Figure 5: Regret (left) and Movement costs (right) on Synthetic Test Benchmarks

Theorems & Definitions (11)

  • Definition 1: Balls in Metric Spaces
  • Definition 2: Minkowski Dimension by Covering
  • Definition 3: Packing Number
  • Remark 4
  • Theorem 5
  • proof
  • Remark 6
  • Corollary 7
  • proof
  • Theorem 8: li2022gaussian
  • ...and 1 more