Predictive Entropy Search for Multi-objective Bayesian Optimization (1511.05467v3)

Abstract: We present PESMO, a Bayesian method for identifying the Pareto set of multi-objective optimization problems, when the functions are expensive to evaluate. The central idea of PESMO is to choose evaluation points so as to maximally reduce the entropy of the posterior distribution over the Pareto set. Critically, the PESMO multi-objective acquisition function can be decomposed as a sum of objective-specific acquisition functions, which enables the algorithm to be used in \emph{decoupled} scenarios in which the objectives can be evaluated separately and perhaps with different costs. This decoupling capability also makes it possible to identify difficult objectives that require more evaluations. PESMO also offers gains in efficiency, as its cost scales linearly with the number of objectives, in comparison to the exponential cost of other methods. We compare PESMO with other related methods for multi-objective Bayesian optimization on synthetic and real-world problems. The results show that PESMO produces better recommendations with a smaller number of evaluations of the objectives, and that a decoupled evaluation can lead to improvements in performance, particularly when the number of objectives is large.

• The paper introduces PESMO, a novel Bayesian method that directly optimizes predictive entropy reduction over the Pareto set for efficient multi-objective optimization.
• PESMO offers computational efficiency, scaling linearly with objectives and demonstrating superior performance on empirical tasks with fewer evaluations compared to other methods.
• The method is practical for expensive function evaluations and lays groundwork for innovations in decoupled, scalable Bayesian optimization, including applications in AutoML and real-time systems.
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Overview of Predictive Entropy Search for Multi-objective Bayesian Optimization

The paper by Daniel Hern andez-Lobato and colleagues on "Predictive Entropy Search for Multi-objective Bayesian Optimization" (PESMO) introduces a novel approach to multi-objective optimization where traditional problems of evaluating multiple real-valued functions are addressed by identifying the Pareto set efficiently. The authors propose a Bayesian method that targets scenarios where evaluating objective functions is expensive and should be minimized to optimize computational resources.

Summary of PESMO

Methodology:

• PESMO revolves around Bayesian optimization principles, emphasizing reduction in predictive entropy over the Pareto set. Unlike many algorithms that transform multi-objective problems into single-objective challenges using scalarization, PESMO directly considers the multi-objective nature, optimizing the entropy reduction across the entire Pareto set.
• The method utilizes Gaussian processes to model uncertainty in objective functions. This is advantageous in the context of functions with no closed form, treating them as black boxes.
• The multi-objective acquisition function in PESMO can be decomposed into a sum of objective-specific acquisition functions, enabling decoupled evaluations where objectives can be evaluated separately based on varying costs.

Technical Contributions:

• PESMO offers a computational efficiency breakthrough, scaling linearly with the number of objectives, distinguishing it from other methods whose computational costs grow exponentially.
• The algorithm identifies objectives that are more challenging, directing more evaluations towards them, effectively managing resources.

Empirical Results

Experiments compared PESMO against other methods on both synthetic and real-world tasks, such as optimizing neural networks on the MNIST dataset. PESMO demonstrated superior capability in generating better recommendations with fewer objective evaluations, especially notable in decoupled scenarios where its independent objective function evaluations lead to performance improvements.

Implications

Practical Effects:

• PESMO is notably beneficial in situations where function evaluations are costly, making it a valuable tool in industries and research areas requiring efficient resource allocation, such as robotic systems optimization and decision-making in financial portfolios.

Theoretical Insights:

• This approach strengthens the role of information theoretic measures, such as entropy reduction, within the field of Bayesian optimization, providing a feasible path forward for tackling high-dimensional, multi-objective problems.

Future Prospects

Looking forward, the development of PESMO lays a foundation for exploring further decoupled evaluation strategies, potentially expanding its reach to complex domains such as automatic machine learning (AutoML) and real-time decision systems in autonomous platforms. Innovations in modular and scalable approaches to Bayesian optimization could be spurred by insights from PESMO, driving research on parallel evaluations and hybrid modeling approaches.