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

Bayesian single- and multi- objective optimisation with nonparametric priors

Optimisation is integral to all sorts of processes in science, economics and arguably underpins the fruition of human intelligence through millions of years of optimisation, or $\textit{evolution}$. Scarce resources make it crucial to maximise their efficient usage. In this thesis, we consider the task of maximising unknown functions which we are able to query point-wise. The function is deemed to be $\textit{costly}$ to evaluate e.g. larger run time or financial expense, requiring a judicious querying strategy given previous observations. We adopt a probabilistic framework for modelling the unknown function and Bayesian non-parametric modelling. In particular, we focus on the $\textit{Gaussian process}$ (GP), a popular non-parametric Bayesian prior on functions. We motivate these choices and give an overview of the Gaussian process in the introduction, and its application to $\textit{Bayesian optimisation}$. A GP's behaviour is intimately controlled by the choice of $\textit{kernel}$ or covariance function, typically chosen to be a parametric function. In chapter 2 we instead place a non-parametric Bayesian prior, known as an Inverse Wishart process prior, over a GP kernel function, and show that this may be marginalised analytically leading to a \textit{Student-t process} (TP). Furthermore we explore a larger class of $\textit{elliptical processes}$, and show that the TP is the most general for which analytic calculation is possible, and apply it successfully for Bayesian optimisation. The remainder of the thesis focusses on various Bayesian optimisation settings. In chapter 3, we consider a setting where we are able to evaluate a function at multiple locations in parallel. Our approach is to consider a measure of information, $\textit{entropy}$, to decide which batch of points to evaluate a function at next. We similarly apply information gain for $\textit{multi-objective}$ Bayesian optimisation in chapter 4. Here, one wishes to find a $\textit{Pareto frontier}$ of efficient settings with respect to several different objectives through sequential evaluation. Finally, in chapter 5 we exploit the idea that in a multi-objective setting, functions are $\textit{correlated}$, incorporating this belief in our choice of prior distribution over the multiple objectives

GPflowOpt: A Bayesian Optimization Library using TensorFlow

A novel Python framework for Bayesian optimization known as GPflowOpt is introduced. The package is based on the popular GPflow library for Gaussian processes, leveraging the benefits of TensorFlow including automatic differentiation, parallelization and GPU computations for Bayesian optimization. Design goals focus on a framework that is easy to extend with custom acquisition functions and models. The framework is thoroughly tested and well documented, and provides scalability. The current released version of GPflowOpt includes some standard single-objective acquisition functions, the state-of-the-art max-value entropy search, as well as a Bayesian multi-objective approach. Finally, it permits easy use of custom modeling strategies implemented in GPflow