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Time-Varying Bayesian Optimization Without a Metronome

Anthony Bardou, Patrick Thiran

TL;DR

This work addresses time-varying Bayesian Optimization when observations impose non-constant sampling frequencies, a regime where GP inference costs grow with data and traditional metronome-based resets can be inefficient. It derives the first regret bound that explicitly depends on the response time $R(n)$ and uses it to advocate principled dataset-size limits and a Wasserstein-distance-based stale-data removal policy. Building on these insights, the authors introduce BOLT, a long-term TVBO algorithm that adapts the dataset size in real time and prunes irrelevant observations, achieving superior performance on diverse synthetic and real-world benchmarks. The results demonstrate that accounting for sampling frequency yields tangible gains in regret and efficiency, with practical impact across domains requiring long-horizon, time-varying optimization.

Abstract

Time-Varying Bayesian Optimization (TVBO) is the go-to framework for optimizing a time-varying, expensive, noisy black-box function $f$. However, most of the asymptotic guarantees offered by TVBO algorithms rely on the assumption that observations are acquired at a constant frequency. As the GP inference complexity scales with the cube of its dataset size, this assumption is unrealistic in the long run. In this paper, we relax this assumption and derive the first upper regret bound that explicitly accounts for changes in the observations sampling frequency. Based on this analysis, we formulate practical recommendations about dataset sizes and stale data policies of TVBO algorithms. We illustrate how an algorithm (BOLT) that follows these recommendations performs better than the state-of-the-art of TVBO through experiments on synthetic and real-world problems.

Time-Varying Bayesian Optimization Without a Metronome

TL;DR

This work addresses time-varying Bayesian Optimization when observations impose non-constant sampling frequencies, a regime where GP inference costs grow with data and traditional metronome-based resets can be inefficient. It derives the first regret bound that explicitly depends on the response time and uses it to advocate principled dataset-size limits and a Wasserstein-distance-based stale-data removal policy. Building on these insights, the authors introduce BOLT, a long-term TVBO algorithm that adapts the dataset size in real time and prunes irrelevant observations, achieving superior performance on diverse synthetic and real-world benchmarks. The results demonstrate that accounting for sampling frequency yields tangible gains in regret and efficiency, with practical impact across domains requiring long-horizon, time-varying optimization.

Abstract

Time-Varying Bayesian Optimization (TVBO) is the go-to framework for optimizing a time-varying, expensive, noisy black-box function . However, most of the asymptotic guarantees offered by TVBO algorithms rely on the assumption that observations are acquired at a constant frequency. As the GP inference complexity scales with the cube of its dataset size, this assumption is unrealistic in the long run. In this paper, we relax this assumption and derive the first upper regret bound that explicitly accounts for changes in the observations sampling frequency. Based on this analysis, we formulate practical recommendations about dataset sizes and stale data policies of TVBO algorithms. We illustrate how an algorithm (BOLT) that follows these recommendations performs better than the state-of-the-art of TVBO through experiments on synthetic and real-world problems.

Paper Structure

This paper contains 32 sections, 5 theorems, 42 equations, 13 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Time-Varying Bayesian Optimization
  4. State-of-the-Art of TVBO
  5. Near-Constant Response Time.
  6. Quasi-Static Problems.
  7. All Other Scenarios.
  8. Main Results
  9. Core Assumptions
  10. Regret Analysis Without a Metronome
  11. Recommendation 1.
  12. Removal of Irrelevant Observations
  13. Recommendation 2.
  14. BOLT: Bayesian Optimization in the Long Term
  15. Estimating the Reponse Time.
  16. ...and 17 more sections

Key Result

Theorem 3.6

Let $\mathcal{A}$ be a TVBO algorithm that uses the GPUCB acquisition function. Let $R(s)$ be the response time of $\mathcal{A}$ for a dataset size $s \in \mathbb{N}$. Let $n$ be the maximal dataset size of $\mathcal{A}$ and let $||\bm u_n||_2^2 = \sum_{i = 1}^n k_T^2(iR(n))$. Pick $\delta \in (0, 1 where $C_1$, $C_2 \in \mathcal{O}(1)$, $\gamma_n \in \mathcal{O}(n)$ and $\beta_T \in \mathcal{O}(\

Figures (13)

  • Figure 1: Recommended dataset size $n^* = \mathop{\mathrm{arg\,max}}\limits_{n \in \mathbb{N}}||\bm u_n||^2_2$. (Left) Recommended dataset size for several common temporal covariance functions $k_T$, under the assumption that the response time is $R(n) \in \Theta(n^3)$. (Right) The recommended dataset size for several response times, under the assumption that $k_T$ is an RBF covariance function.
  • Figure 2: Normalized average regret across the benchmarks (lower is better). For each benchmark, the best performing TVBO algorithm gets a normalized regret of 0, and the worst performing TVBO algorithm gets a normalized regret of 1. The normalized regrets are then averaged across all the benchmarks.
  • Figure 3: Evolution of the dataset sizes $n$ of the TVBO algorithms on the Eggholder (left) and Powell (right) synthetic functions. The plots are in log scale.
  • Figure 4: (Left) Average response time and average regrets of the TVBO solutions during the optimization of the Schwefel synthetic function. (Right) Dataset sizes of the TVBO solutions during the optimization of the Schwefel synthetic function.
  • Figure 5: (Left) Average response time and average regrets of the TVBO solutions during the optimization of the Eggholder synthetic function. (Right) Dataset sizes of the TVBO solutions during the optimization of the Eggholder synthetic function.
  • ...and 8 more figures

Theorems & Definitions (11)

  • Definition 3.5: Response Time
  • Theorem 3.6
  • Theorem 3.8
  • Lemma A.1
  • proof
  • Lemma A.2
  • proof
  • Lemma A.3
  • proof
  • proof
  • ...and 1 more