Max-value Entropy Search for Efficient Bayesian Optimization (1703.01968v3)

Abstract: Entropy Search (ES) and Predictive Entropy Search (PES) are popular and empirically successful Bayesian Optimization techniques. Both rely on a compelling information-theoretic motivation, and maximize the information gained about the $\arg\max$ of the unknown function; yet, both are plagued by the expensive computation for estimating entropies. We propose a new criterion, Max-value Entropy Search (MES), that instead uses the information about the maximum function value. We show relations of MES to other Bayesian optimization methods, and establish a regret bound. We observe that MES maintains or improves the good empirical performance of ES/PES, while tremendously lightening the computational burden. In particular, MES is much more robust to the number of samples used for computing the entropy, and hence more efficient for higher dimensional problems.

• The paper presents MES, a novel method that directly estimates the maximum value, reducing complexity compared to traditional entropy search methods.
• The paper establishes a regret bound for MES, providing theoretical backing along with empirical validation on synthetic, benchmark, and real-world tasks.
• The paper demonstrates MES's scalability and adaptability, using additive Gaussian processes to extend Bayesian optimization to high-dimensional problems.

Max-value Entropy Search for Efficient Bayesian Optimization

This paper addresses the computational challenges inherent in Bayesian optimization (BO), particularly those associated with techniques such as Entropy Search (ES) and Predictive Entropy Search (PES), which rely on information gain about the location of the maximum of an unknown black-box function. These methods, while empirically effective, suffer from prohibitive computational costs due to the necessity of entropy estimation in high-dimensional spaces. The authors introduce a novel method, Max-value Entropy Search (MES), which focuses on estimating the maximum value instead of its corresponding argument. This shift from $\arg\max$ to the maximum value reduces computational complexity significantly.

Key advancements presented include the development of MES, detailed analysis of its connections to existing BO strategies, and theoretical insights such as a regret bound. The paper's contributions are multifaceted and positioned to enhance BO's applicability across varied high-dimensional problem domains:

1. Reduction in Computational Complexity: MES mitigates the complexity seen in ES and PES. Instead of sampling over the entire input space, MES involves a simple one-dimensional maximum value estimation. This makes MES more scalable and suitable for problems of higher dimensionality.
2. Regret Bound: The authors establish a regret bound for MES, marking a notable achievement as no such bounds existed for previous entropy-based methods. This theoretical result supports the empirical performance improvements demonstrated through MES.
3. Empirical Validations: Through various empirical evaluations, MES is shown to perform on par or better than PES and ES while operating much faster. The evaluations span synthetic functions, optimization benchmarks, and real-world tasks such as neural network hyperparameter tuning and robotic action learning.
4. High-dimensional Extension via Additive GPs: The paper demonstrates MES extended to high-dimensional problems using additive Gaussian processes (GPs). This approach leverages potential additivity in model structures to maintain efficiency in optimization procedures.

In practical implications, MES offers a substantial leap forward for applications in robotics, machine learning, and other engineering fields where optimization of non-convex, expensive functions is required. Given the scalability improvements, it allows for quicker iterations and convergence towards optimal solutions, thereby reducing computational resource requirements and enabling real-time decision-making in practice.

The comparisons to other BO techniques such as GP-UCB, PI, EI, as well as recent ES methods provide a strong contextual grounding for researchers looking to choose an appropriate BO method based on their specific problem's complexity and computational constraints. The comparisons further highlight the modular nature of MES, enabling it to be easily adapted or extended, suggesting areas for further exploration such as leveraging recent advances in deep neural networks for better feature extraction for BO.

Future research directions might focus on improving the robustness of the Gumbel approximation, analyzing cases with mixed discrete-continuous variable spaces, and exploring further integration of MES with modern machine learning frameworks to enhance its applicability to emerging domains, such as automated machine learning (AutoML) pipelines and industrial design optimization.

Overall, this paper offers significant insights and tools for tackling high-dimensional Bayesian optimization challenges, pushing the boundaries of what can be efficiently achieved under computational constraints. The future of MES looks promising, especially in contexts demanding rapid, high-efficiency BO applications.