Improving sample efficiency of high dimensional Bayesian optimization with MCMC
(2401.02650)Abstract
Sequential optimization methods are often confronted with the curse of dimensionality in high-dimensional spaces. Current approaches under the Gaussian process framework are still burdened by the computational complexity of tracking Gaussian process posteriors and need to partition the optimization problem into small regions to ensure exploration or assume an underlying low-dimensional structure. With the idea of transiting the candidate points towards more promising positions, we propose a new method based on Markov Chain Monte Carlo to efficiently sample from an approximated posterior. We provide theoretical guarantees of its convergence in the Gaussian process Thompson sampling setting. We also show experimentally that both the Metropolis-Hastings and the Langevin Dynamics version of our algorithm outperform state-of-the-art methods in high-dimensional sequential optimization and reinforcement learning benchmarks.
Overview
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Bayesian optimization (BO) is a strategy for optimizing functions without gradient information, often using Gaussian processes (GPs), but struggles with high-dimensional spaces.
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The paper introduces MCMC-BO, an algorithm combining BO with Markov Chain Monte Carlo techniques to improve sample efficiency in high-dimensional optimization.
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MCMC-BO transitions candidate points towards promising areas based on Gaussian process Thompson sampling without the need for a large discretized mesh, reducing computational load.
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Theoretical guarantees are provided for the convergence of MCMC-BO, suggesting that it manages the trade-off between exploration and exploitation efficiently in high dimensions.
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Experiments demonstrate MCMC-BO's superior performance over traditional BO methods, maintaining effectiveness as problem dimensions increase.
Introduction
Bayesian optimization (BO) is an effective approach for optimizing black-box functions that lack gradient information. This approach has seen success across various real-world engineering challenges and machine learning applications like hyper-parameter tuning. At the heart of BO lies a surrogate model that helps make intelligent guesses about the objective function's landscape, with Gaussian processes (GPs) being a typical choice for this model due to their probabilistic nature and ability to manage uncertainty.
However, as the problem's dimensionality increases, BO falls prey to the curse of dimensionality, leading to computational inefficiencies and a potential explosion in the candidate sample space. This paper introduces a novel approach, named MCMC-BO, that uses Markov Chain Monte Carlo (MCMC) techniques to enhance the sample efficiency for doing BO in high-dimensional spaces.
Related Work
Existing efforts to mitigate the dimensionality problem in BO have focused on creating partitions in the search space, ranging from trust regions to tree-based partitions that isolate promising regions for sampling. Although these methods improve sampling strategies, they often rely on discretizing the search space, a technique that becomes less effective as dimensionality grows. As a result, the challenge remains to balance exploring the vast uncertainty with exploiting known good regions, all while keeping computational demands in check.
Algorithm Design
At its core, MCMC-BO is an algorithm that integrates the principles of BO with the sampling prowess of MCMC. The procedure transitions candidate points towards areas of the search space that show promise based on Gaussian process Thompson sampling. Unlike traditional BO, which may require storing and computing over a vast discretized mesh of points, MCMC-BO only tracks a manageable batch of points. By doing so, the algorithm retains theoretical performance guarantees while significantly reducing memory and computational overhead.
The paper details the implementation of two MCMC strategies: Metropolis-Hastings (MH) and Langevin Dynamics (LD). Both these methods are adapted to the context of BO and are used to transit candidate points in accordance with the Gaussian process model's posterior.
Theoretical Guarantee and Experiments
The authors provide a theoretical framework that guarantees the convergence of MCMC-BO. They postulate that the algorithm can effectively circumvent the limitations posed by high dimensionality, striking a balance between exploration and exploitation without the excessive memory use traditionally associated with fine-grain discretization of high-dimensional spaces.
Experimental evidence showcases the superiority of MCMC-BO over standard BO methods and state-of-the-art high-dimensional BO strategies across various benchmarks, including challenging Mujoco locomotion tasks. Notably, the experiments demonstrate that MCMC-BO maintains good performance even as problem dimensions scale.
Conclusion
MCMC-BO addresses the critical challenge of sample efficiency in high-dimensional Bayesian optimization by introducing an MCMC-based local optimization method. Its theoretical foundations ensure that performance is not compromised even as dimensionality scales. As optimization tasks in high-dimensional spaces become more prevalent, methodologies like MCMC-BO will become increasingly valuable. The authors highlight that there is room for further enhancements, particularly in parallel computing and analytical backward computations, offering an exciting avenue for ongoing research.
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