Bayesian Optimization has become the reference method for the global optimization of black box, expensive and possibly noisy functions. Bayesian Op-timization learns a probabilistic model about the objective function, usually a Gaussian Process, and builds, depending on its mean and variance, an acquisition function whose optimizer yields the new evaluation point, leading to update the probabilistic surrogate model. Despite its sample efficiency, Bayesian Optimiza-tion does not scale well with the dimensions of the problem. The optimization of the acquisition function has received less attention because its computational cost is usually considered negligible compared to that of the evaluation of the objec-tive function. Its efficient optimization is often inhibited, particularly in high di-mensional problems, by multiple extrema. In this paper we leverage the addition-ality of the objective function into mapping both the kernel and the acquisition function of the Bayesian Optimization in lower dimensional subspaces. This ap-proach makes more efficient the learning/updating of the probabilistic surrogate model and allows an efficient optimization of the acquisition function. Experi-mental results are presented for real-life application, that is the control of pumps in urban water distribution systems.
Imperial College London（帝国理工学院）；University of Freiburg（弗赖堡大学）
Bayesian optimization is a sample-efficient approach to global optimization that relies on theoretically motivated value heuristics (acquisition functions) to guide its search process. Fully maximizing acquisition functions produces the Bayes' decision rule, but this ideal is difficult to achieve since these functions are frequently non-trivial to optimize. This statement is especially true when evaluating queries in parallel, where acquisition functions are routinely non-convex, high-dimensional, and intractable. We first show that acquisition functions estimated via Monte Carlo integration a…
College of Engineering, Pune（浦那工程学院）；National Institute of Technology, Srinagar（斯利那加国家理工学院）
Bayesian Optimization is an effective method for searching the global maxima of an objective function especially if the function is unknown. The process comprises of using a surrogate function and choosing an acquisition function followed by optimizing the acquisition function to find the next sampling point. This paper analyzes different acquistion functions like Maximum Probability of Improvement and Expected Improvement and various optimizers like L-BFGS and TNC to optimize the acquisitions functions for finding the next sampling point. Along with the analysis of time taken, the paper also …
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…
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…
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…
Huawei R&D UK（华为英国研发部）；University of Cambridge（剑桥大学）；University College London（伦敦大学学院）
Bayesian optimisation presents a sample-efficient methodology for global optimisation. Within this framework, a crucial performance-determining subroutine is the maximisation of the acquisition function, a task complicated by the fact that acquisition functions tend to be non-convex and thus nontrivial to optimise. In this paper, we undertake a comprehensive empirical study of approaches to maximise the acquisition function. Additionally, by deriving novel, yet mathematically equivalent, compositional forms for popular acquisition functions, we recast the maximisation task as a compositional o…
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 …
Fraunhofer Center for Machine Learning（弗劳恩霍夫机器学习中心）；Fraunhofer Institute for Intelligent Analysis and Information Systems IAIS（弗劳恩霍夫智能分析与信息系统研究所）
A solution that is only reliable under favourable conditions is hardly a safe solution. Min Max Optimization is an approach that returns optima that are robust against worst case conditions. We propose algorithms that perform Min Max Optimization in a setting where the function that should be optimized is not known a priori and hence has to be learned by experiments. Therefore we extend the Bayesian Optimization setting, which is tailored to maximization problems, to Min Max Optimization problems. While related work extends the two acquisition functions Expected Improvement and Gaussian Proces…
Pohang University of Science and Technology（浦项科技大学）；BARO AI（巴罗人工智能公司）
Bayesian optimization is a sample-efficient method for finding a global optimum of an expensive-to-evaluate black-box function. A global solution is found by accumulating a pair of query point and its function value, repeating these two procedures: (i) modeling a surrogate function; (ii) maximizing an acquisition function to determine where next to query. Convergence guarantees are only valid when the global optimizer of the acquisition function is found at each round and selected as the next query point. In practice, however, local optimizers of an acquisition function are also used, since se…
Brigham Young University（杨百翰大学）；Harvard University（哈佛大学）
This paper introduces a probabilistic framework to estimate parameters of an acquisition function given observed human behavior that can be modeled as a collection of sample paths from a Bayesian optimization procedure. The methodology involves defining a likelihood on observed human behavior from an optimization task, where the likelihood is parameterized by a Bayesian optimization subroutine governed by an unknown acquisition function. This structure enables us to make inference on a subject's acquisition function while allowing their behavior to deviate around the solution to the Bayesian o…
Bayesian Optimization is the state of the art technique for the optimization of black boxes, i.e., functions where we do not have access to their analytical expression nor its gradients, they are expensive to evaluate and its evaluation is noisy. The most popular application of bayesian optimization is the automatic hyperparameter tuning of machine learning algorithms, where we obtain the best configuration of machine learning algorithms by optimizing the estimation of the generalization error of these algorithms. Despite being applied with success, bayesian optimization methodologies also hav…
Bayesian optimization (BO) is a popular method to optimize expensive black-box functions. It efficiently tunes machine learning algorithms under the implicit assumption that hyperparameter evaluations cost approximately the same. In reality, the cost of evaluating different hyperparameters, be it in terms of time, dollars or energy, can span several orders of magnitude of difference. While a number of heuristics have been proposed to make BO cost-aware, none of these have been proven to work robustly. In this work, we reformulate cost-aware BO in terms of Pareto efficiency and introduce the co…
TNO（荷兰国家应用科学研究院）；University of Exeter（埃克塞特大学）；Swansea University（斯旺西大学）；Delft University of Technology（代尔夫特理工大学）
Performing multi-objective Bayesian optimisation by scalarising the objectives avoids the computation of expensive multi-dimensional integral-based acquisition functions, instead of allowing one-dimensional standard acquisition functions\textemdash such as Expected Improvement\textemdash to be applied. Here, two infill criteria based on hypervolume improvement\textemdash one recently introduced and one novel\textemdash are compared with the multi-surrogate Expected Hypervolume Improvement. The reasons for the disparities in these methods' effectiveness in maximising the hypervolume of the acqu…
Lund University（隆德大学）；University of Freiburg（弗赖堡大学）；Federal University of Minas Gerais（米纳斯吉拉斯联邦大学）；Leibniz University Hannover（汉诺威莱布尼茨大学）；Bosch Center for Artificial Intelligence（博世人工智能中心）；Stanford University（斯坦福大学）
Bayesian optimization (BO) has become an established framework and popular tool for hyperparameter optimization (HPO) of machine learning (ML) algorithms. While known for its sample-efficiency, vanilla BO can not utilize readily available prior beliefs the practitioner has on the potential location of the optimum. Thus, BO disregards a valuable source of information, reducing its appeal to ML practitioners. To address this issue, we propose $π$BO, an acquisition function generalization which incorporates prior beliefs about the location of the optimum in the form of a probability distribution,…
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…
In many real-world scenarios where data is high dimensional, test time acquisition of features is a non-trivial task due to costs associated with feature acquisition and evaluating feature value. The need for highly confident models with an extremely frugal acquisition of features can be addressed by allowing a feature selection method to become target aware. We introduce an approach to feature selection that is based on Bayesian learning, allowing us to report target-specific levels of uncertainty, false positive, and false negative rates. In addition, measuring uncertainty lifts the restrict…
Bayesian optimization is a class of data efficient model based algorithms typically focused on global optimization. We consider the more general case where a user is faced with multiple problems that each need to be optimized conditional on a state variable, for example given a range of cities with different patient distributions, we optimize the ambulance locations conditioned on patient distribution. Given partitions of CIFAR-10, we optimize CNN hyperparameters for each partition. Similarity across objectives boosts optimization of each objective in two ways: in modelling by data sharing acr…
Nanjing University of Aeronautics and Astronautics（南京航空航天大学）；Collaborative Innovation Center of Novel Software Technology and Industrialization（新型软件技术与产业化协同创新中心）；State Key Laboratory of Aerodynamics（空气动力学国家重点实验室）
In Bayesian optimization (BO) for expensive black-box optimization tasks, acquisition function (AF) guides sequential sampling and plays a pivotal role for efficient convergence to better optima. Prevailing AFs usually rely on artificial experiences in terms of preferences for exploration or exploitation, which runs a risk of a computational waste or traps in local optima and resultant re-optimization. To address the crux, the idea of data-driven AF selection is proposed, and the sequential AF selection task is further formalized as a Markov decision process (MDP) and resort to powerful reinfo…
Bayesian optimization (BO) is a class of sample-efficient global optimization methods, where a probabilistic model conditioned on previous observations is used to determine future evaluations via the optimization of an acquisition function. Most acquisition functions are myopic, meaning that they only consider the impact of the next function evaluation. Non-myopic acquisition functions consider the impact of the next $h$ function evaluations and are typically computed through rollout, in which $h$ steps of BO are simulated. These rollout acquisition functions are defined as $h$-dimensional int…
German Center for Neurodegenerative Diseases (DZNE)（德国神经退行性疾病中心（DZNE））；University of Glasgow（格拉斯哥大学）
Bayesian optimization (BO) is a popular algorithm for solving challenging optimization tasks. It is designed for problems where the objective function is expensive to evaluate, perhaps not available in exact form, without gradient information and possibly returning noisy values. Different versions of the algorithm vary in the choice of the acquisition function, which recommends the point to query the objective at next. Initially, researchers focused on improvement-based acquisitions, while recently the attention has shifted to more computationally expensive information-theoretical measures. In…
BOOST: A Data-Driven Framework for the Automated Joint Selection of Kernel and Acquisition Functions in Bayesian Optimization
Department of Chemical and Biomolecular Engineering, Korea Advanced Institute of Science and Technology（化学与生物分子工程系，韩国科学技术院）；Department of AX, Korea Advanced Institute of Science and Technology（AX系，韩国科学技术院）；Saudi Aramco-KAIST CO2 Management Center（沙特阿美-KAIST二氧化碳管理中心）
We propose a novel Bayesian method to solve the maximization of a time-dependent expensive-to-evaluate stochastic oracle. We are interested in the decision that maximizes the oracle at a finite time horizon, given a limited budget of noisy evaluations of the oracle that can be performed before the horizon. Our recursive two-step lookahead acquisition function for Bayesian optimization makes nonmyopic decisions at every stage by maximizing the expected utility at the specified time horizon. Specifically, we propose a generalized two-step lookahead framework with a customizable \emph{value} func…
NTT Communication Science Laboratories（NTT通信科学实验室）
For Bayesian optimization (BO) on high-dimensional data with complex structure, neural network-based kernels for Gaussian processes (GPs) have been used to learn flexible surrogate functions by the high representation power of deep learning. However, existing methods train neural networks by maximizing the marginal likelihood, which do not directly improve the BO performance. In this paper, we propose a meta-learning method for BO with neural network-based kernels that minimizes the expected gap between the true optimum value and the best value found by BO. We model a policy, which takes the c…
We propose MUMBO, the first high-performing yet computationally efficient acquisition function for multi-task Bayesian optimization. Here, the challenge is to perform efficient optimization by evaluating low-cost functions somehow related to our true target function. This is a broad class of problems including the popular task of multi-fidelity optimization. However, while information-theoretic acquisition functions are known to provide state-of-the-art Bayesian optimization, existing implementations for multi-task scenarios have prohibitive computational requirements. Previous acquisition fun…
University of British Columbia（不列颠哥伦比亚大学）；University of Oxford（牛津大学）；University of Cambridge（剑桥大学）；Canadian Institute for Advanced Research（加拿大高等研究院）
Bayesian optimization is a sample-efficient method for black-box global optimization. How- ever, the performance of a Bayesian optimization method very much depends on its exploration strategy, i.e. the choice of acquisition function, and it is not clear a priori which choice will result in superior performance. While portfolio methods provide an effective, principled way of combining a collection of acquisition functions, they are often based on measures of past performance which can be misleading. To address this issue, we introduce the Entropy Search Portfolio (ESP): a novel approach to por…
Nanjing University（南京大学）；University of Science and Technology of China（中国科学技术大学）
Bayesian optimization (BO) is a popular approach for expensive black-box optimization, with applications including parameter tuning, experimental design, robotics. BO usually models the objective function by a Gaussian process (GP), and iteratively samples the next data point by maximizing an acquisition function. In this paper, we propose a new general framework for BO by generating pseudo-points (i.e., data points whose objective values are not evaluated) to improve the GP model. With the classic acquisition function, i.e., upper confidence bound (UCB), we prove that the cumulative regret ca…
We propose a novel Bayesian method to solve the maximization of a time-dependent expensive-to-evaluate oracle. We are interested in the decision that maximizes the oracle at a finite time horizon, when relatively few noisy evaluations can be performed before the horizon. Our recursive, two-step lookahead expected payoff ($\texttt{r2LEY}$) acquisition function makes nonmyopic decisions at every stage by maximizing the estimated expected value of the oracle at the horizon. $\texttt{r2LEY}$ circumvents the evaluation of the expensive multistep (more than two steps) lookahead acquisition function …
University of Oxford（牛津大学）；University of British Columbia（不列颠哥伦比亚大学）；Rocket Gaming Systems
Bayesian optimization is a powerful global optimization technique for expensive black-box functions. One of its shortcomings is that it requires auxiliary optimization of an acquisition function at each iteration. This auxiliary optimization can be costly and very hard to carry out in practice. Moreover, it creates serious theoretical concerns, as most of the convergence results assume that the exact optimum of the acquisition function can be found. In this paper, we introduce a new technique for efficient global optimization that combines Gaussian process confidence bounds and treed simultane…
Bayesian optimization with Gaussian processes has become an increasingly popular tool in the machine learning community. It is efficient and can be used when very little is known about the objective function, making it popular in expensive black-box optimization scenarios. It uses Bayesian methods to sample the objective efficiently using an acquisition function which incorporates the model's estimate of the objective and the uncertainty at any given point. However, there are several different parameterized acquisition functions in the literature, and it is often unclear which one to use. Inst…
Kobe Steel, Ltd.（神户制钢所）；The Institute of Statistical Mathematics（统计数理研究所）；RIKEN AIP（理化学研究所AIP）
Bayesian optimization is an effective method to efficiently optimize unknown objective functions with high evaluation costs. Traditional Bayesian optimization algorithms select one point per iteration for single objective function, whereas in recent years, Bayesian optimization for multi-objective optimization or multi-point search per iteration have been proposed. However, Bayesian optimization that can deal with them at the same time in non-heuristic way is not known at present. We propose a Bayesian optimization algorithm that can deal with multi-objective optimization and multi-point searc…
ALINE: Joint Amortization for Bayesian Inference and Active Data Acquisition
ELLIS Institute Finland（芬兰ELLIS研究所）；Department of Computer Science, Aalto University, Finland（芬兰阿尔托大学计算机科学系）；Department of Computer Science, University of Helsinki, Finland（芬兰赫尔辛基大学计算机科学系）；Department of Computer Science, University of Manchester, UK（英国曼彻斯特大学计算机科学系）
Recent advances in computationally efficient non-myopic Bayesian optimization (BO) improve query efficiency over traditional myopic methods like expected improvement while only modestly increasing computational cost. These advances have been largely limited, however, to unconstrained optimization. For constrained optimization, the few existing non-myopic BO methods require heavy computation. For instance, one existing non-myopic constrained BO method [Lam and Willcox, 2017] relies on computationally expensive unreliable brute-force derivative-free optimization of a Monte Carlo rollout acquisit…
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…
Bayesian optimization is an approach to optimizing objective functions that take a long time (minutes or hours) to evaluate. It is best-suited for optimization over continuous domains of less than 20 dimensions, and tolerates stochastic noise in function evaluations. It builds a surrogate for the objective and quantifies the uncertainty in that surrogate using a Bayesian machine learning technique, Gaussian process regression, and then uses an acquisition function defined from this surrogate to decide where to sample. In this tutorial, we describe how Bayesian optimization works, including Gau…
Bayesian Optimization (BO) is a framework for black-box optimization that is especially suitable for expensive cost functions. Among the main parts of a BO algorithm, the acquisition function is of fundamental importance, since it guides the optimization algorithm by translating the uncertainty of the regression model in a utility measure for each point to be evaluated. Considering such aspect, selection and design of acquisition functions are one of the most popular research topics in BO. Since no single acquisition function was proved to have better performance in all tasks, a well-establish…
Sandia National Laboratories（桑迪亚国家实验室）；Xilinx Inc.（赛灵思公司）；Medpace Inc.（Medpace公司）；GIW Industries Inc.（GIW工业公司）
High-fidelity complex engineering simulations are highly predictive, but also computationally expensive and often require substantial computational efforts. The mitigation of computational burden is usually enabled through parallelism in high-performance cluster (HPC) architecture. In this paper, an asynchronous constrained batch-parallel Bayesian optimization method is proposed to efficiently solve the computationally-expensive simulation-based optimization problems on the HPC platform, with a budgeted computational resource, where the maximum number of simulations is a constant. The advantag…
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…
Bayesian Optimization of Function Networks with Partial Evaluations
Center for Applied Mathematics, Cornell University（康奈尔大学应用数学中心）；School of Operations Research and Information Engineering, Cornell University（康奈尔大学运筹学与信息工程学院）；Department of Computing and Mathematical Sciences, Caltech（加州理工学院计算与数学科学系）；Meta
Skoltech（斯科尔科沃科技学院）；National Research University Higher School of Economics（俄罗斯国立高等经济研究大学）；IITP RAS（俄罗斯科学院信息传输问题研究所）
The key idea of Bayesian optimization is replacing an expensive target function with a cheap surrogate model. By selection of an acquisition function for Bayesian optimization, we trade off between exploration and exploitation. The acquisition function typically depends on the mean and the variance of the surrogate model at a given point. The most common Gaussian process-based surrogate model assumes that the target with fixed parameters is a realization of a Gaussian process. However, often the target function doesn't satisfy this approximation. Here we consider target functions that come fro…
ETH Zürich（苏黎世联邦理工学院）；Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；Bosch Center for Artificial Intelligence（博世人工智能中心）；University of Freiburg（弗莱堡大学）
Transferring knowledge across tasks to improve data-efficiency is one of the open key challenges in the field of global black-box optimization. Readily available algorithms are typically designed to be universal optimizers and, therefore, often suboptimal for specific tasks. We propose a novel transfer learning method to obtain customized optimizers within the well-established framework of Bayesian optimization, allowing our algorithm to utilize the proven generalization capabilities of Gaussian processes. Using reinforcement learning to meta-train an acquisition function (AF) on a set of rela…
The acquisition function, a critical component in Bayesian optimization (BO), can often be written as the expectation of a utility function under a surrogate model. However, to ensure that acquisition functions are tractable to optimize, restrictions must be placed on the surrogate model and utility function. To extend BO to a broader class of models and utilities, we propose likelihood-free BO (LFBO), an approach based on likelihood-free inference. LFBO directly models the acquisition function without having to separately perform inference with a probabilistic surrogate model. We show that co…
