Efficient nonmyopic Bayesian optimization and quadrature

Finite-horizon sequential decision problems arise naturally in many machine learning contexts, including Bayesian optimization and Bayesian quadrature. Computing the optimal policy for such problems requires solving Bellman equations, which are generally intractable. Most existing work resorts to myopic approximations by limiting the decision horizon to only a single time-step, which can perform poorly at balancing exploration and exploitation. We propose a general framework for efficient, nonmyopic approximation of the optimal policy by drawing a connection between the optimal adaptive policy and its non-adaptive counterpart. Our proposal is to compute an optimal batch of points, then select a single point from within this batch to evaluate. We realize this idea for both Bayesian optimization and Bayesian quadrature and demonstrate that our proposed method significantly outperforms common myopic alternatives on a variety of tasks.

Automated Model Selection with Bayesian Quadrature

We present a novel technique for tailoring Bayesian quadrature (BQ) to model selection. The state-of-the-art for comparing the evidence of multiple models relies on Monte Carlo methods, which converge slowly and are unreliable for computationally expensive models. Previous research has shown that BQ offers sample efficiency superior to Monte Carlo in computing the evidence of an individual model. However, applying BQ directly to model comparison may waste computation producing an overly-accurate estimate for the evidence of a clearly poor model. We propose an automated and efficient algorithm for computing the most-relevant quantity for model selection: the posterior probability of a model. Our technique maximizes the mutual information between this quantity and observations of the models' likelihoods, yielding efficient acquisition of samples across disparate model spaces when likelihood observations are limited. Our method produces more-accurate model posterior estimates using fewer model likelihood evaluations than standard Bayesian quadrature and Monte Carlo estimators, as we demonstrate on synthetic and real-world examples.

Improving Quadrature for Constrained Integrands

We present an improved Bayesian framework for performing inference of affine transformations of constrained functions. We focus on quadrature with nonnegative functions, a common task in Bayesian inference. We consider constraints on the range of the function of interest, such as nonnegativity or boundedness. Although our framework is general, we derive explicit approximation schemes for these constraints, and argue for the use of a log transformation for functions with high dynamic range such as likelihood surfaces. We propose a novel method for optimizing hyperparameters in this framework: we optimize the marginal likelihood in the original space, as opposed to in the transformed space. The result is a model that better explains the actual data. Experiments on synthetic and real-world data demonstrate our framework achieves superior estimates using less wall-clock time than existing Bayesian quadrature procedures.