Most research in Bayesian optimization (BO) has focused on \emph{direct feedback} scenarios, where one has access to exact values of some expensive-to-evaluate objective. This direction has been mainly driven by the use of BO in machine learning hyper-parameter configuration problems. However, in domains such as modelling human preferences, A/B tests, or recommender systems, there is a need for methods that can replace direct feedback with \emph{preferential feedback}, obtained via rankings or pairwise comparisons. In this work, we present preferential batch Bayesian optimization (PBBO), a new framework that allows finding the optimum of a latent function of interest, given any type of parallel preferential feedback for a group of two or more points. We do so by using a Gaussian process model with a likelihood specially designed to enable parallel and efficient data collection mechanisms, which are key in modern machine learning. We show how the acquisitions developed under this framework generalize and augment previous approaches in Bayesian optimization, expanding the use of these techniques to a wider range of domains. An extensive simulation study shows the benefits of this approach, both with simulated functions and four real data sets.
Helsinki Institute for Information Technology HIIT（赫尔辛基信息技术研究所）；Aalto University（阿尔托大学）；The University of Manchester（曼彻斯特大学）
Bayesian optimization is an effective method for finding extrema of a black-box function. We propose a new type of Bayesian optimization for learning user preferences in high-dimensional spaces. The central assumption is that the underlying objective function cannot be evaluated directly, but instead a minimizer along a projection can be queried, which we call a projective preferential query. The form of the query allows for feedback that is natural for a human to give, and which enables interaction. This is demonstrated in a user experiment in which the user feedback comes in the form of opti…
We present two algorithms for Bayesian optimization in the batch feedback setting, based on Gaussian process upper confidence bound and Thompson sampling approaches, along with frequentist regret guarantees and numerical results.
We develop BatchBALD, a tractable approximation to the mutual information between a batch of points and model parameters, which we use as an acquisition function to select multiple informative points jointly for the task of deep Bayesian active learning. BatchBALD is a greedy linear-time $1 - \frac{1}{e}$-approximate algorithm amenable to dynamic programming and efficient caching. We compare BatchBALD to the commonly used approach for batch data acquisition and find that the current approach acquires similar and redundant points, sometimes performing worse than randomly acquiring data. We fini…
In Bayesian Optimization (BO) we study black-box function optimization with noisy point evaluations and Bayesian priors. Convergence of BO can be greatly sped up by batching, where multiple evaluations of the black-box function are performed in a single round. The main difficulty in this setting is to propose at the same time diverse and informative batches of evaluation points. In this work, we introduce DPP-Batch Bayesian Optimization (DPP-BBO), a universal framework for inducing batch diversity in sampling based BO by leveraging the repulsive properties of Determinantal Point Processes (DPP…
University of Amsterdam（阿姆斯特丹大学）；Qualcomm AI Research（高通人工智能研究院）
In this work we propose a batch Bayesian optimization method for combinatorial problems on permutations, which is well suited for expensive-to-evaluate objectives. We first introduce LAW, an efficient batch acquisition method based on determinantal point processes using the acquisition weighted kernel. Relying on multiple parallel evaluations, LAW enables accelerated search on combinatorial spaces. We then apply the framework to permutation problems, which have so far received little attention in the Bayesian Optimization literature, despite their practical importance. We call this method LAW2…
Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究院）；Technology Innovation Institute（技术创新研究院）
In deep active learning, it is especially important to choose multiple examples to markup at each step to work efficiently, especially on large datasets. At the same time, existing solutions to this problem in the Bayesian setup, such as BatchBALD, have significant limitations in selecting a large number of examples, associated with the exponential complexity of computing mutual information for joint random variables. We, therefore, present the Large BatchBALD algorithm, which gives a well-grounded approximation to the BatchBALD method that aims to achieve comparable quality while being more c…
Preferential Bayesian optimization (PBO) is a framework for optimizing a decision maker's latent utility function using preference feedback. This work introduces the expected utility of the best option (qEUBO) as a novel acquisition function for PBO. When the decision maker's responses are noise-free, we show that qEUBO is one-step Bayes optimal and thus equivalent to the popular knowledge gradient acquisition function. We also show that qEUBO enjoys an additive constant approximation guarantee to the one-step Bayes-optimal policy when the decision maker's responses are corrupted by noise. We …
Institute for Data Science in Mechanical Engineering, RWTH Aachen University（机械工程数据科学研究所，亚琛工业大学）；Department of Clinical Research, University of Bern（伯尔尼大学临床研究系）；Center for Reproducible Science and Research Synthesis, University of Zurich（苏黎世大学可重复科学与研究综合中心）；aiXopt GmbH（aiXopt公司）
Imperial College London（伦敦帝国理工学院）；University of Manchester（曼彻斯特大学）；University of Edinburgh（爱丁堡大学）；Alan Turing Institute（艾伦·图灵研究所）
Bayesian Optimisation (BO) methods seek to find global optima of objective functions which are only available as a black-box or are expensive to evaluate. Such methods construct a surrogate model for the objective function, quantifying the uncertainty in that surrogate through Bayesian inference. Objective evaluations are sequentially determined by maximising an acquisition function at each step. However, this ancilliary optimisation problem can be highly non-trivial to solve, due to the non-convexity of the acquisition function, particularly in the case of batch Bayesian optimisation, where m…
University of Oxford（牛津大学）；UC Berkeley（加州大学伯克利分校）；IBM Research, Zurich（IBM研究院苏黎世分部）
We present Acquisition Thompson Sampling (ATS), a novel technique for batch Bayesian Optimization (BO) based on the idea of sampling multiple acquisition functions from a stochastic process. We define this process through the dependency of the acquisition functions on a set of model hyper-parameters. ATS is conceptually simple, straightforward to implement and, unlike other batch BO methods, it can be employed to parallelize any sequential acquisition function or to make existing parallel methods scale further. We present experiments on a variety of benchmark functions and on the hyper-paramet…
Bayesian optimization (BO) and its batch extensions are successful for optimizing expensive black-box functions. However, these traditional BO approaches are not yet ideal for optimizing less expensive functions when the computational cost of BO can dominate the cost of evaluating the blackbox function. Examples of these less expensive functions are cheap machine learning models, inexpensive physical experiment through simulators, and acquisition function optimization in Bayesian optimization. In this paper, we consider a batch BO setting for situations where function evaluations are less expe…
Leveraging the wealth of unlabeled data produced in recent years provides great potential for improving supervised models. When the cost of acquiring labels is high, probabilistic active learning methods can be used to greedily select the most informative data points to be labeled. However, for many large-scale problems standard greedy procedures become computationally infeasible and suffer from negligible model change. In this paper, we introduce a novel Bayesian batch active learning approach that mitigates these issues. Our approach is motivated by approximating the complete data posterior …
Bayesian optimization (BO) is a powerful approach for optimizing black-box, expensive-to-evaluate functions. To enable a flexible trade-off between the cost and accuracy, many applications allow the function to be evaluated at different fidelities. In order to reduce the optimization cost while maximizing the benefit-cost ratio, in this paper, we propose Batch Multi-fidelity Bayesian Optimization with Deep Auto-Regressive Networks (BMBO-DARN). We use a set of Bayesian neural networks to construct a fully auto-regressive model, which is expressive enough to capture strong yet complex relationsh…
Integration over non-negative integrands is a central problem in machine learning (e.g. for model averaging, (hyper-)parameter marginalisation, and computing posterior predictive distributions). Bayesian Quadrature is a probabilistic numerical integration technique that performs promisingly when compared to traditional Markov Chain Monte Carlo methods. However, in contrast to easily-parallelised MCMC methods, Bayesian Quadrature methods have, thus far, been essentially serial in nature, selecting a single point to sample at each step of the algorithm. We deliver methods to select batches of po…
We present K-Means Batch Bayesian Optimization (KMBBO), a novel batch sampling algorithm for Bayesian Optimization (BO). KMBBO uses unsupervised learning to efficiently estimate peaks of the model acquisition function. We show in empirical experiments that our method outperforms the current state-of-the-art batch allocation algorithms on a variety of test problems including tuning of algorithm hyper-parameters and a challenging drug discovery problem. In order to accommodate the real-world problem of high dimensional data, we propose a modification to KMBBO by combining it with compressed sens…
Sorbonne Université（索邦大学）；INSERM（法国国家健康与医学研究院）；CNRS（法国国家科学研究中心）；Institut de la Vision（视觉研究所）
Bayesian optimization (BO) is an effective approach to optimize expensive black-box functions, that seeks to trade-off between exploitation (selecting parameters where the maximum is likely) and exploration (selecting parameters where we are uncertain about the objective function). In many real-world situations, direct measurements of the objective function are not possible, and only binary measurements such as success/failure or pairwise comparisons are available. To perform efficient exploration in this setting, we show that it is important for BO algorithms to distinguish between different …
In many applications of black-box optimization, one can evaluate multiple points simultaneously, e.g. when evaluating the performances of several different neural network architectures in a parallel computing environment. In this paper, we develop a novel batch Bayesian optimization algorithm --- the parallel knowledge gradient method. By construction, this method provides the one-step Bayes-optimal batch of points to sample. We provide an efficient strategy for computing this Bayes-optimal batch of points, and we demonstrate that the parallel knowledge gradient method finds global optima sign…
CERMICS, Ecole des Ponts（巴黎桥路学校 CERMICS 实验室）；Inria Paris（法国国家信息与自动化研究所巴黎中心）
Bayesian inference allows to obtain useful information on the parameters of models, either in computational statistics or more recently in the context of Bayesian Neural Networks. The computational cost of usual Monte Carlo methods for sampling posterior laws in Bayesian inference scales linearly with the number of data points. One option to reduce it to a fraction of this cost is to resort to mini-batching in conjunction with unadjusted discretizations of Langevin dynamics, in which case only a random fraction of the data is used to estimate the gradient. However, this leads to an additional …
Bayesian Optimization is a useful tool for experiment design. Unfortunately, the classical, sequential setting of Bayesian Optimization does not translate well into laboratory experiments, for instance battery design, where measurements may come from different sources and their evaluations may require significant waiting times. Multi-fidelity Bayesian Optimization addresses the setting with measurements from different sources. Asynchronous batch Bayesian Optimization provides a framework to select new experiments before the results of the prior experiments are revealed. This paper proposes an …
University of Oxford（牛津大学）；Mind Foundry Ltd.（Mind Foundry有限公司）；Oxford-Man Institute of Quantitative Finance（牛津-曼定量金融研究所）
Batch Bayesian optimisation (BO) has been successfully applied to hyperparameter tuning using parallel computing, but it is wasteful of resources: workers that complete jobs ahead of others are left idle. We address this problem by developing an approach, Penalising Locally for Asynchronous Bayesian Optimisation on $k$ workers (PLAyBOOK), for asynchronous parallel BO. We demonstrate empirically the efficacy of PLAyBOOK and its variants on synthetic tasks and a real-world problem. We undertake a comparison between synchronous and asynchronous BO, and show that asynchronous BO often outperforms …
We show that training a deep network using batch normalization is equivalent to approximate inference in Bayesian models. We further demonstrate that this finding allows us to make meaningful estimates of the model uncertainty using conventional architectures, without modifications to the network or the training procedure. Our approach is thoroughly validated by measuring the quality of uncertainty in a series of empirical experiments on different tasks. It outperforms baselines with strong statistical significance, and displays competitive performance with recent Bayesian approaches.
Department of Computer Science, University of Helsinki（赫尔辛基大学计算机科学系）；Department of Signal Theory and Communications, Universidad Carlos III de Madrid（马德里卡洛斯三世大学信号理论与通信系）；VTT Technical Research Centre of Finland（芬兰技术研究中心）
Graduate School of Information Science and Technology（信息科学技术研究生院）；The University of Tokyo（东京大学）；NTT Communication Science Laboratories（NTT通信科学实验室）；Epistra Inc.（Epistra公司）
The optimization of high-dimensional black-box functions is a challenging problem. When a low-dimensional linear embedding structure can be assumed, existing Bayesian optimization (BO) methods often transform the original problem into optimization in a low-dimensional space. They exploit the low-dimensional structure and reduce the computational burden. However, we reveal that this approach could be limited or inefficient in exploring the high-dimensional space mainly due to the biased reconstruction of the high-dimensional queries from the low-dimensional queries. In this paper, we investigat…
Center for Pattern Recognition and Data Analytics（模式识别与数据分析中心）；Deakin University（迪肯大学）
Parameter settings profoundly impact the performance of machine learning algorithms and laboratory experiments. The classical grid search or trial-error methods are exponentially expensive in large parameter spaces, and Bayesian optimization (BO) offers an elegant alternative for global optimization of black box functions. In situations where the black box function can be evaluated at multiple points simultaneously, batch Bayesian optimization is used. Current batch BO approaches are restrictive in that they fix the number of evaluations per batch, and this can be wasteful when the number of s…
Trinity College（都柏林圣三一学院）；Dalle Molle Institute for Artificial Intelligence Research (IDSIA) - USI/SUPSI（达勒莫尔人工智能研究所（IDSIA）- 瑞士意大利语区应用艺术与科学大学/瑞士南部应用科学与艺术大学）
Preferential Bayesian optimisation (PBO) deals with optimisation problems where the objective function can only be accessed via preference judgments, such as "this is better than that" between two candidate solutions (like in A/B tests or recommender systems). The state-of-the-art approach to PBO uses a Gaussian process to model the preference function and a Bernoulli likelihood to model the observed pairwise comparisons. Laplace's method is then employed to compute posterior inferences and, in particular, to build an appropriate acquisition function. In this paper, we prove that the true post…
Bayesian optimisation (BO) algorithms have shown remarkable success in applications involving expensive black-box functions. Traditionally BO has been set as a sequential decision-making process which estimates the utility of query points via an acquisition function and a prior over functions, such as a Gaussian process. Recently, however, a reformulation of BO via density-ratio estimation (BORE) allowed reinterpreting the acquisition function as a probabilistic binary classifier, removing the need for an explicit prior over functions and increasing scalability. In this paper, we present a the…
Gaussian processes (GPs) are the main surrogate functions used for sequential modelling such as Bayesian Optimization and Active Learning. Their drawbacks are poor scaling with data and the need to run an optimization loop when using a non-Gaussian likelihood. In this paper, we focus on `fantasizing' batch acquisition functions that need the ability to condition on new fantasized data computationally efficiently. By using a sparse Dual GP parameterization, we gain linear scaling with batch size as well as one-step updates for non-Gaussian likelihoods, thus extending sparse models to greedy bat…
Bayesian optimization (BO) has emerged during the last few years as an effective approach to optimizing black-box functions where direct queries of the objective are expensive. In this paper we consider the case where direct access to the function is not possible, but information about user preferences is. Such scenarios arise in problems where human preferences are modeled, such as A/B tests or recommender systems. We present a new framework for this scenario that we call Preferential Bayesian Optimization (PBO) which allows us to find the optimum of a latent function that can only be queried…
This paper describes a general-purpose extension of max-value entropy search, a popular approach for Bayesian Optimisation (BO). A novel approximation is proposed for the information gain -- an information-theoretic quantity central to solving a range of BO problems, including noisy, multi-fidelity and batch optimisations across both continuous and highly-structured discrete spaces. Previously, these problems have been tackled separately within information-theoretic BO, each requiring a different sophisticated approximation scheme, except for batch BO, for which no computationally-lightweight …
We develop parallel predictive entropy search (PPES), a novel algorithm for Bayesian optimization of expensive black-box objective functions. At each iteration, PPES aims to select a batch of points which will maximize the information gain about the global maximizer of the objective. Well known strategies exist for suggesting a single evaluation point based on previous observations, while far fewer are known for selecting batches of points to evaluate in parallel. The few batch selection schemes that have been studied all resort to greedy methods to compute an optimal batch. To the best of our…
In many high-throughput experimental design settings, such as those common in biochemical engineering, batched queries are more cost effective than one-by-one sequential queries. Furthermore, it is often not possible to directly choose items to query. Instead, the experimenter specifies a set of constraints that generates a library of possible items, which are then selected stochastically. Motivated by these considerations, we investigate \emph{Batched Stochastic Bayesian Optimization} (BSBO), a novel Bayesian optimization scheme for choosing the constraints in order to guide exploration towar…
This paper re-examines the problem of parameter estimation in Bayesian networks with missing values and hidden variables from the perspective of recent work in on-line learning [Kivinen & Warmuth, 1994]. We provide a unified framework for parameter estimation that encompasses both on-line learning, where the model is continuously adapted to new data cases as they arrive, and the more traditional batch learning, where a pre-accumulated set of samples is used in a one-time model selection process. In the batch case, our framework encompasses both the gradient projection algorithm and the EM algo…
Massachusetts Institute of Technology（麻省理工学院）；DeepMind（深度思考公司）
Optimization of high-dimensional black-box functions is an extremely challenging problem. While Bayesian optimization has emerged as a popular approach for optimizing black-box functions, its applicability has been limited to low-dimensional problems due to its computational and statistical challenges arising from high-dimensional settings. In this paper, we propose to tackle these challenges by (1) assuming a latent additive structure in the function and inferring it properly for more efficient and effective BO, and (2) performing multiple evaluations in parallel to reduce the number of itera…
Protein sequence design is a challenging problem in protein engineering, which aims to discover novel proteins with useful biological functions. Directed evolution is a widely-used approach for protein sequence design, which mimics the evolution cycle in a laboratory environment and conducts an iterative protocol. However, the burden of laboratory experiments can be reduced by using machine learning approaches to build a surrogate model of the protein landscape and conducting in-silico population selection through model-based fitness prediction. In this paper, we propose a new method based on …
In this work we investigate the reasons why Batch Normalization (BN) improves the generalization performance of deep networks. We argue that one major reason, distinguishing it from data-independent normalization methods, is randomness of batch statistics. This randomness appears in the parameters rather than in activations and admits an interpretation as a practical Bayesian learning. We apply this idea to other (deterministic) normalization techniques that are oblivious to the batch size. We show that their generalization performance can be improved significantly by Bayesian learning of the …
Bayesian optimization (BO) has become an effective approach for black-box function optimization problems when function evaluations are expensive and the optimum can be achieved within a relatively small number of queries. However, many cases, such as the ones with high-dimensional inputs, may require a much larger number of observations for optimization. Despite an abundance of observations thanks to parallel experiments, current BO techniques have been limited to merely a few thousand observations. In this paper, we propose ensemble Bayesian optimization (EBO) to address three current challen…
University of Exeter（埃克塞特大学）；Swansea University（斯旺西大学）
Bayesian optimisation is a popular, surrogate model-based approach for optimising expensive black-box functions. Given a surrogate model, the next location to expensively evaluate is chosen via maximisation of a cheap-to-query acquisition function. We present an $ε$-greedy procedure for Bayesian optimisation in batch settings in which the black-box function can be evaluated multiple times in parallel. Our $ε$-shotgun algorithm leverages the model's prediction, uncertainty, and the approximated rate of change of the landscape to determine the spread of batch solutions to be distributed around a…
We propose a novel, theoretically-grounded, acquisition function for Batch Bayesian optimization informed by insights from distributionally ambiguous optimization. Our acquisition function is a lower bound on the well-known Expected Improvement function, which requires evaluation of a Gaussian Expectation over a multivariate piecewise affine function. Our bound is computed instead by evaluating the best-case expectation over all probability distributions consistent with the same mean and variance as the original Gaussian distribution. Unlike alternative approaches, including Expected Improveme…
Oregon State University（俄勒冈州立大学）；University of Texas at Austin（德克萨斯大学奥斯汀分校）
Bayesian Optimization aims at optimizing an unknown non-convex/concave function that is costly to evaluate. We are interested in application scenarios where concurrent function evaluations are possible. Under such a setting, BO could choose to either sequentially evaluate the function, one input at a time and wait for the output of the function before making the next selection, or evaluate the function at a batch of multiple inputs at once. These two different settings are commonly referred to as the sequential and batch settings of Bayesian Optimization. In general, the sequential setting lea…
National University of Singapore（新加坡国立大学）；MIT（麻省理工学院）
This paper presents a novel approach to top-$k$ ranking Bayesian optimization (top-$k$ ranking BO) which is a practical and significant generalization of preferential BO to handle top-$k$ ranking and tie/indifference observations. We first design a surrogate model that is not only capable of catering to the above observations, but is also supported by a classic random utility model. Another equally important contribution is the introduction of the first information-theoretic acquisition function in BO with preferential observation called multinomial predictive entropy search (MPES) which is fl…
Oregon State University（俄勒冈州立大学）；University of Texas at Austin（德克萨斯大学奥斯汀分校）
Bayesian optimization (BO) algorithms try to optimize an unknown function that is expensive to evaluate using minimum number of evaluations/experiments. Most of the proposed algorithms in BO are sequential, where only one experiment is selected at each iteration. This method can be time inefficient when each experiment takes a long time and more than one experiment can be ran concurrently. On the other hand, requesting a fix-sized batch of experiments at each iteration causes performance inefficiency in BO compared to the sequential policies. In this paper, we present an algorithm that asks a …
