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High-dimensional Bayesian Optimization via Covariance Matrix Adaptation Strategy

Lam Ngo, Huong Ha, Jeffrey Chan, Vu Nguyen, Hongyu Zhang

TL;DR

This work tackles the challenge of applying Bayesian Optimization in high-dimensional settings by introducing a Covariance Matrix Adaptation Strategy (CMA) as a meta-algorithm to define local regions likely to contain the global optimum. The CMA-based meta-algorithm learns a Gaussian search distribution p(x)=N(m,σ^2C) that concentrates around promising regions, constructing an α-level hyper-ellipsoid around the CMA mean to guide where BO should search, and integrating with standard BO, TuRBO, or BAxUS. Through three instantiated variants—CMA-BO, CMA-TuRBO, and CMA-BAxUS—the approach demonstrates improved data efficiency and robust performance across synthetic and real-world high-dimensional benchmarks, often outperforming state-of-the-art optimizers and meta-algorithms. The results suggest CMA-driven local region modeling can effectively mitigate the curse of dimensionality in BO and offer practical, scalable improvements for expensive black-box optimization tasks.

Abstract

Bayesian Optimization (BO) is an effective method for finding the global optimum of expensive black-box functions. However, it is well known that applying BO to high-dimensional optimization problems is challenging. To address this issue, a promising solution is to use a local search strategy that partitions the search domain into local regions with high likelihood of containing the global optimum, and then use BO to optimize the objective function within these regions. In this paper, we propose a novel technique for defining the local regions using the Covariance Matrix Adaptation (CMA) strategy. Specifically, we use CMA to learn a search distribution that can estimate the probabilities of data points being the global optimum of the objective function. Based on this search distribution, we then define the local regions consisting of data points with high probabilities of being the global optimum. Our approach serves as a meta-algorithm as it can incorporate existing black-box BO optimizers, such as BO, TuRBO, and BAxUS, to find the global optimum of the objective function within our derived local regions. We evaluate our proposed method on various benchmark synthetic and real-world problems. The results demonstrate that our method outperforms existing state-of-the-art techniques.

High-dimensional Bayesian Optimization via Covariance Matrix Adaptation Strategy

TL;DR

This work tackles the challenge of applying Bayesian Optimization in high-dimensional settings by introducing a Covariance Matrix Adaptation Strategy (CMA) as a meta-algorithm to define local regions likely to contain the global optimum. The CMA-based meta-algorithm learns a Gaussian search distribution p(x)=N(m,σ^2C) that concentrates around promising regions, constructing an α-level hyper-ellipsoid around the CMA mean to guide where BO should search, and integrating with standard BO, TuRBO, or BAxUS. Through three instantiated variants—CMA-BO, CMA-TuRBO, and CMA-BAxUS—the approach demonstrates improved data efficiency and robust performance across synthetic and real-world high-dimensional benchmarks, often outperforming state-of-the-art optimizers and meta-algorithms. The results suggest CMA-driven local region modeling can effectively mitigate the curse of dimensionality in BO and offer practical, scalable improvements for expensive black-box optimization tasks.

Abstract

Bayesian Optimization (BO) is an effective method for finding the global optimum of expensive black-box functions. However, it is well known that applying BO to high-dimensional optimization problems is challenging. To address this issue, a promising solution is to use a local search strategy that partitions the search domain into local regions with high likelihood of containing the global optimum, and then use BO to optimize the objective function within these regions. In this paper, we propose a novel technique for defining the local regions using the Covariance Matrix Adaptation (CMA) strategy. Specifically, we use CMA to learn a search distribution that can estimate the probabilities of data points being the global optimum of the objective function. Based on this search distribution, we then define the local regions consisting of data points with high probabilities of being the global optimum. Our approach serves as a meta-algorithm as it can incorporate existing black-box BO optimizers, such as BO, TuRBO, and BAxUS, to find the global optimum of the objective function within our derived local regions. We evaluate our proposed method on various benchmark synthetic and real-world problems. The results demonstrate that our method outperforms existing state-of-the-art techniques.
Paper Structure (62 sections, 15 equations, 17 figures, 1 table, 4 algorithms)

This paper contains 62 sections, 15 equations, 17 figures, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Bayesian Optimization
  4. The Thompson Sampling Acquisition Function
  5. TurBO
  6. BAxUS
  7. Related Work
  8. High-dimensional Bayesian Optimization via Covariance Matrix Adaptation
  9. The Covariance Matrix Adaptation Strategy
  10. The CMA-based Meta-algorithm
  11. Overall Process.
  12. Local Region Formulation.
  13. Local Optimization.
  14. Restart Strategy.
  15. CMA-BO: CMA-based Meta-algorithm with Standard BO
  16. ...and 47 more sections

Figures (17)

  • Figure 1: Illustration of the proposed CMA-based meta-algorithm. In step (1), a hyper-ellipsoid local region is initialized. In step (2), a BO optimizer (e.g., BO, TuRBO, BAxUS) is used in this local region to collect a population of candidates (data points to be evaluated). In step (3), the local region is updated using the CMA technique. The process is conducted iteratively until the evaluation budget is depleted.
  • Figure 2: Comparison between the CMA-based BO methods (CMA-BO, CMA-TuRBO, CMA-BAxUS) against the original BO optimizers (BO, TuRBO, BAxUS). Plotting the mean and standard error over 10 repetitions. The CMA-based BO methods outperform their respective BO optimizers in most cases.
  • Figure 3: Comparison between the CMA-based BO methods (CMA-BO, CMA-TuRBO) against existing meta-algorithms (LAMCTS-TuRBO, MCTSVS-BO, MCTSVS-TuRBO). Plotting the mean and standard error over 10 repetitions. The CMA-based BO methods outperform the other meta-algorithms given the same BO optimizer.
  • Figure 4: Comparison between the CMA-based BO methods (CMA-BO, CMA-TuRBO, CMA-BAxUS) against the CMA-based ES methods (CMA-ES, DTS-CMAES) and BADS, a global optimization method which combines BO and CMA-ES. Plotting the mean and standard error over 10 repetitions. The CMA-based BO methods outperform CMA-ES, DTS-CMAES and BADS consistently.
  • Figure 5: The local regions' trajectories defined by the proposed CMA-based meta-algorithm when paired with BO (upper row) and TuRBO (lower row) for the Shifted-Alpine-2D function. In this case, the local regions of CMA-BO gradually move towards the global minimum of the objective function whilst the local regions of CMA-TuRBO quickly converge to a sub-optimal location, then restart and move toward to the global optimum.
  • ...and 12 more figures