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EntropySearch (ES) [15],Predictive Entropy Search(PES)[16]and the earlier work of IAGO [46]select queries by maximizing this quantity.

Specifically,at each iteration the black-box function is evaluated at one or more input locations that maximizes anacquisition function onthemodel.

Information-theoretic Bayesian optimization techniques have become popular for optimizing expensive-to-evaluate black-box functions due to their non-myopic qualities. Entropy Search and Predictive Entropy Search both consider the entropy over the optimum in the input space, while the recent Max-value Entropy Search considers the entropy over the optimal value in the output space. We propose Joint Entropy Search (JES), a novel information-theoretic acquisition function that considers an entirely new quantity, namely the entropy over the joint optimal probability density over both input and output space. To incorporate this information, we consider the reduction in entropy from conditioning on fantasized optimal input/output pairs. The resulting approach primarily relies on standard GP machinery and removes complex approximations typically associated with information-theoretic methods. With minimal computational overhead, JES shows superior decision-making, and yields state-of-the-art performance for information-theoretic approaches across a wide suite of tasks. As a light-weight approach with superior results, JES provides a new go-to acquisition function for Bayesian optimization.

Searching large and complex design spaces for a global optimum can be infeasible and unnecessary. A practical alternative is to iteratively refine the neighborhood of an initial design using local optimization methods such as gradient descent. We propose local entropy search (LES), a Bayesian optimization paradigm that explicitly targets the solutions reachable by the descent sequences of iterative optimizers. The algorithm propagates the posterior belief over the objective through the optimizer, resulting in a probability distribution over descent sequences. It then selects the next evaluation by maximizing mutual information with that distribution, using a combination of analytic entropy calculations and Monte-Carlo sampling of descent sequences. Empirical results on high-complexity synthetic objectives and benchmark problems show that LES achieves strong sample efficiency compared to existing local and global Bayesian optimization methods.

For example, in hardware design optimization, we need to find the designs that trade-off performance, energy, and area overhead using expensive computational simulations.

First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. The paper under review'Predictive Entropy Search for Efficient Global Optimization of Black-box Functions' presents an approach for Bayesian optimization that measures entropy in the posterior predictive distribution. The authors present a method named Predictive Entropy Search (PES) that is derived by a reparameterization of the expected information gain based on the symmetry of mutual information. The authors describe the posterior sampling, predictive entropy approximation and hyper-parameter learning steps of PES. The advantages of the proposed method are that it is more efficient and accurate than alternatives and that it allows to handle the hyper-parameters in a fully Bayesian way.

A bilevel optimization problem consists of two optimization problems nested as an upper- and a lower-level problem, in which the optimality of the lower-level problem defines a constraint for the upper-level problem. This paper considers Bayesian optimization (BO) for the case that both the upper- and lower-levels involve expensive black-box functions. Because of its nested structure, bilevel optimization has a complex problem definition and, compared with other standard extensions of BO such as multi-objective or constraint settings, it has not been widely studied. We propose an information-theoretic approach that considers the information gain of both the upper- and lower-optimal solutions and values. This enables us to define a unified criterion that measures the benefit for both level problems, simultaneously. Further, we also show a practical lower bound based approach to evaluating the information gain. We empirically demonstrate the effectiveness of our proposed method through several benchmark datasets.

Entropy Search and Predictive Entropy Search both consider the entropy over the optimum in the input space, while the recent Max-value Entropy Search considers the entropy over the optimal value in the output space.