Batch Bayesian Optimization via Local Penalization

The popularity of Bayesian optimization methods for efficient exploration of parameter spaces has lead to a series of papers applying Gaussian processes as surrogates in the optimization of functions. However, most proposed approaches only allow the exploration of the parameter space to occur sequentially. Often, it is desirable to simultaneously propose batches of parameter values to explore. This is particularly the case when large parallel processing facilities are available. These facilities could be computational or physical facets of the process being optimized. E.g. in biological experiments many experimental set ups allow several samples to be simultaneously processed. Batch methods, however, require modeling of the interaction between the evaluations in the batch, which can be expensive in complex scenarios. We investigate a simple heuristic based on an estimate of the Lipschitz constant that captures the most important aspect of this interaction (i.e. local repulsion) at negligible computational overhead. The resulting algorithm compares well, in running time, with much more elaborate alternatives. The approach assumes that the function of interest, $f$, is a Lipschitz continuous function. A wrap-loop around the acquisition function is used to collect batches of points of certain size minimizing the non-parallelizable computational effort. The speed-up of our method with respect to previous approaches is significant in a set of computationally expensive experiments.Comment: 11 pages, 10 figure

Achieving robustness to aleatoric uncertainty with heteroscedastic Bayesian optimisation

Bayesian optimisation is a sample-efficient search methodology that holds great promise for accelerating drug and materials discovery programs. A frequently-overlooked modelling consideration in Bayesian optimisation strategies however, is the representation of heteroscedastic aleatoric uncertainty. In many practical applications it is desirable to identify inputs with low aleatoric noise, an example of which might be a material composition which consistently displays robust properties in response to a noisy fabrication process. In this paper, we propose a heteroscedastic Bayesian optimisation scheme capable of representing and minimising aleatoric noise across the input space. Our scheme employs a heteroscedastic Gaussian process (GP) surrogate model in conjunction with two straightforward adaptations of existing acquisition functions. First, we extend the augmented expected improvement (AEI) heuristic to the heteroscedastic setting and second, we introduce the aleatoric noise-penalised expected improvement (ANPEI) heuristic. Both methodologies are capable of penalising aleatoric noise in the suggestions and yield improved performance relative to homoscedastic Bayesian optimisation and random sampling on toy problems as well as on two real-world scientific datasets. Code is available at: \url{https://github.com/Ryan-Rhys/Heteroscedastic-BO

Treed Gaussian Process Regression for Solving Offline Data-Driven Continuous Multiobjective Optimization Problems

This is the final version. Available from MIT Press via the DOI in this recordFor offline data-driven multiobjective optimization problems (MOPs), no new data is available during the optimization process. Approximation models (or surrogates) are first built using the provided offline data and an optimizer, e.g. a multiobjective evolutionary algorithm, can then be utilized to find Pareto optimal solutions to the problem with surrogates as objective functions. In contrast to online data-driven MOPs, these surrogates cannot be updated with new data and, hence, the approximation accuracy cannot be improved by considering new data during the optimization process. Gaussian process regression (GPR) models are widely used as surrogates because of their ability to provide uncertainty information. However, building GPRs becomes computationally expensive when the size of the dataset is large. Using sparse GPRs reduces the computational cost of building the surrogates. However, sparse GPRs are not tailored to solve offline data-driven MOPs, where good accuracy of the surrogates is needed near Pareto optimal solutions. Treed GPR (TGPR-MO) surrogates for offline data-driven MOPs with continuous decision variables are proposed in this paper. The proposed surrogates first split the decision space into subregions using regression trees and build GPRs sequentially in regions close to Pareto optimal solutions in the decision space to accurately approximate tradeoffs between the objective functions. TGPR-MO surrogates are computationally inexpensive because GPRs are built only in a smaller region of the decision space utilizing a subset of the data. The TGPR-MO surrogates were tested on distance-based visualizable problems with various data sizes, sampling strategies, numbers of objective functions, and decision variables. Experimental results showed that the TGPR-MO surrogates are computationally cheaper and can handle datasets of large size. Furthermore, TGPR-MO surrogates produced solutions closer to Pareto optimal solutions compared to full GPRs and sparse GPRs.Academy of Finlan

Robust expected improvement for Bayesian optimization

Bayesian Optimization (BO) links Gaussian Process (GP) surrogates with sequential design toward optimizing expensive-to-evaluate black-box functions. Example design heuristics, or so-called acquisition functions, like expected improvement (EI), balance exploration and exploitation to furnish global solutions under stringent evaluation budgets. However, they fall short when solving for robust optima, meaning a preference for solutions in a wider domain of attraction. Robust solutions are useful when inputs are imprecisely specified, or where a series of solutions is desired. A common mathematical programming technique in such settings involves an adversarial objective, biasing a local solver away from ``sharp'' troughs. Here we propose a surrogate modeling and active learning technique called robust expected improvement (REI) that ports adversarial methodology into the BO/GP framework. After describing the methods, we illustrate and draw comparisons to several competitors on benchmark synthetic exercises and real problems of varying complexity.Comment: 27 pages, 17 figures, 1 tabl

Information Directed Sampling and Bandits with Heteroscedastic Noise

In the stochastic bandit problem, the goal is to maximize an unknown function via a sequence of noisy evaluations. Typically, the observation noise is assumed to be independent of the evaluation point and to satisfy a tail bound uniformly on the domain; a restrictive assumption for many applications. In this work, we consider bandits with heteroscedastic noise, where we explicitly allow the noise distribution to depend on the evaluation point. We show that this leads to new trade-offs for information and regret, which are not taken into account by existing approaches like upper confidence bound algorithms (UCB) or Thompson Sampling. To address these shortcomings, we introduce a frequentist regret analysis framework, that is similar to the Bayesian framework of Russo and Van Roy (2014), and we prove a new high-probability regret bound for general, possibly randomized policies, which depends on a quantity we refer to as regret-information ratio. From this bound, we define a frequentist version of Information Directed Sampling (IDS) to minimize the regret-information ratio over all possible action sampling distributions. This further relies on concentration inequalities for online least squares regression in separable Hilbert spaces, which we generalize to the case of heteroscedastic noise. We then formulate several variants of IDS for linear and reproducing kernel Hilbert space response functions, yielding novel algorithms for Bayesian optimization. We also prove frequentist regret bounds, which in the homoscedastic case recover known bounds for UCB, but can be much better when the noise is heteroscedastic. Empirically, we demonstrate in a linear setting with heteroscedastic noise, that some of our methods can outperform UCB and Thompson Sampling, while staying competitive when the noise is homoscedastic.Comment: Figure 1a,2a update