We present a tutorial on Bayesian optimization, a method of finding the maximum of expensive cost functions. Bayesian optimization employs the Bayesian technique of setting a prior over the objective function and combining it with evidence to get a posterior function. This permits a utility-based selection of the next observation to make on the objective function, which must take into account both exploration (sampling from areas of high uncertainty) and exploitation (sampling areas likely to offer improvement over the current best observation). We also present two detailed extensions of Bayesian optimization, with experiments-active user modelling with preferences, and hierarchical reinforcement learning-and a discussion of the pros and cons of Bayesian optimization based on our experiences.
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Bayesian Optimisation (BO) is a technique used in optimising a$D$-dimensional function which is typically expensive to evaluate. While therehave been many successes for BO in low dimensions, scaling it to highdimensions has been notoriously difficult. Existing literature on the topic areunder very restrictive settings. In this paper, we identify two key challengesin this endeavour. We tackle these challenges by assuming an additive structurefor the function. This setting is substantially more expressive and contains aricher class of functions than previous work. We prove that, for additivefunctions the regret has only linear dependence on $D$ even though the functiondepends on all $D$ dimensions. We also demonstrate several other statisticaland computational benefits in our framework. Via synthetic examples, ascientific simulation and a face detection problem we demonstrate that ourmethod outperforms naive BO on additive functions and on several examples wherethe function is not additive.
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最近,人们越来越关注贝叶斯优化 - 一种未知函数的优化,其假设通常由高斯过程(GP)先前表示。我们研究了一种直接使用函数argmax估计的优化策略。该策略提供了实践和理论上的优势:不需要选择权衡参数,而且,我们建立与流行的GP-UCB和GP-PI策略的紧密联系。我们的方法可以被理解为自动和自适应地在GP-UCB和GP-PI中进行勘探和利用。我们通过对遗憾的界限以及对机器人和视觉任务的广泛经验评估来说明这种自适应调整的效果,展示了该策略对一系列性能标准的稳健性。
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贝叶斯优化和Lipschitz优化已经开发出用于优化黑盒功能的替代技术。它们各自利用关于函数的不同形式的先验。在这项工作中,我们探索了这些技术的策略,以便更好地进行全局优化。特别是,我们提出了在传统BO算法中使用Lipschitz连续性假设的方法,我们称之为Lipschitz贝叶斯优化(LBO)。这种方法不会增加渐近运行时间,并且在某些情况下会大大提高性能(而在最坏的情况下,性能类似)。实际上,在一个特定的环境中,我们证明使用Lipschitz信息产生与后悔相同或更好的界限,而不是单独使用贝叶斯优化。此外,我们提出了一个简单的启发式方法来估计Lipschitz常数,并证明Lipschitz常数的增长估计在某种意义上是“无害的”。我们对具有4个采集函数的15个数据集进行的实验表明,在最坏的情况下,LBO的表现类似于底层BO方法,而在某些情况下,它的表现要好得多。特别是汤普森采样通常看到了极大的改进(因为Lipschitz信息已经得到了很好的修正) - 探索“现象”及其LBO变体通常优于其他采集功能。
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How can we take advantage of opportunities for experimental parallelization in exploration-exploitation tradeoffs? In many experimental scenarios, it is often desirable to execute experiments simultaneously or in batches, rather than only performing one at a time. Additionally , observations may be both noisy and expensive. We introduce Gaussian Process Batch Upper Confidence Bound (GP-BUCB), an upper confidence bound-based algorithm, which models the reward function as a sample from a Gaussian process and which can select batches of experiments to run in parallel. We prove a general regret bound for GP-BUCB, as well as the surprising result that for some common kernels, the asymptotic average regret can be made independent of the batch size. The GP-BUCB algorithm is also applicable in the related case of a delay between initiation of an experiment and observation of its results , for which the same regret bounds hold. We also introduce Gaussian Process Adaptive Upper Confidence Bound (GP-AUCB), a variant of GP-BUCB which can exploit parallelism in an adaptive manner. We evaluate GP-BUCB and GP-AUCB on several simulated and real data sets. These experiments show that GP-BUCB and GP-AUCB are competitive with state-of-the-art heuristics. 1
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Bayesian optimization is a sample-efficient method for black-box global optimization. However , 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 portfolio construction which is motivated by information theoretic considerations. We show that ESP outperforms existing portfolio methods on several real and synthetic problems, including geostatistical datasets and simulated control tasks. We not only show that ESP is able to offer performance as good as the best, but unknown, acquisition function, but surprisingly it often gives better performance. Finally , over a wide range of conditions we find that ESP is robust to the inclusion of poor acquisition functions.
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In this paper, we analyze a generic algorithm scheme for sequential global optimization using Gaussian processes. The upper bounds we derive on the cumulative regret for this generic algorithm improve by an exponential factor the previously known bounds for algorithms like GP-UCB. We also introduce the novel Gaussian Process Mutual Information algorithm (GP-MI), which significantly improves further these upper bounds for the cumulative regret. We confirm the efficiency of this algorithm on synthetic and real tasks against the natural competitor, GP-UCB, and also the Expected Improvement heuristic.
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贝叶斯优化通常假设给出贝叶斯先验。然而,贝叶斯优化中强有力的理论保证在实践中经常因为先验中的未知参数而受到损害。在本文中,我们采用经验贝叶斯的变量并表明,通过估计从同一个先前采样的离线数据之前的高斯过程和构建后验的无偏估计,GP-UCB的变体和改进概率实现近乎零的后悔界限,其随着离线数据和离线数据的数量减少到与观测噪声成比例的常数。在线评估的数量增加。根据经验,我们已经验证了我们的方法,以挑战模拟机器人问题为特色的任务和运动规划。
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许多应用程序需要优化评估成本昂贵的未知噪声函数。我们将这项任务正式化为一个多臂强盗问题,其中支付函数要么是从高斯过程(GP)中采样,要么是具有低RKHS范数。我们解决了导致此设置后悔限制的重要开放问题,这意味着GP优化的新收敛速度。 Weanalyze GP-UCB,一种直观的基于上置信度的算法,并且在最大信息增益方面限制了它的累积遗憾,在GP优化和实验设计之间建立了新的连接。此外,根据运算符光谱对后者进行处理,我们获得了许多常用协方差函数的显式次线性区域边界。在一些重要的案例中,我们的界限对维度的依赖程度令人惊讶。在我们对真实传感器数据的实验中,GP-UCB与其他的GP优化方法相比具有优势。
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机器学习的许多实际应用需要数据有效的黑盒功能优化,例如,识别超参数或过程设置。然而,容易获得的算法通常被设计为通用优化器,因此对于特定任务而言通常是次优的。因此,提出了一种学习优化器的方法,该优化器自动适应于给定类别的目标函数,例如,在sim-to-realapplications的上下文中。所提出的方法不是从头开始学习优化,而是基于着名的贝叶斯优化框架。只有采集函数(AF)被学习的神经网络所取代,因此得到的算法仍然能够利用高斯过程的经过验证的广义化能力。我们在几个模拟以及模拟到真实传输任务上进行实验。结果表明,学习的优化器(1)在一般函数类上始终表现优于或与已知AF相媲美,并且(2)可以使用廉价模拟自动识别函数类的结构属性并转换该知识以快速适应实际硬件任务,从而显着优于现有的与问题无关的AF。
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基于高斯过程模型的贝叶斯优化(BO)是优化评估成本昂贵的黑盒函数的有力范例。虽然几个BO算法可证明地收敛到未知函数的全局最优,但他们认为内核的超参数是已知的。在实践中情况并非如此,并且错误指定经常导致这些算法收敛到较差的局部最优。在本文中,我们提出了第一个BO算法,它可以证明是无后悔的,并且在不参考超参数的情况下收敛到最优。我们慢慢地调整了固定核的超参数,从而扩展了相关的函数类超时,使BO算法考虑了更复杂的函数候选。基于理论上的见解,我们提出了几种实用的算法,通过在线超参数估计来实现BO的经验数据效率,但是保留理论收敛保证。我们评估了几个基准问题的方法。
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We propose minimum regret search (MRS), a novel acquisition function for Bayesian optimization. MRS bears similarities with information-theoretic approaches such as en-tropy search (ES). However, while ES aims in each query at maximizing the information gain with respect to the global maximum, MRS aims at minimizing the expected simple regret of its ultimate recommendation for the optimum. While empirically ES and MRS perform similar in most of the cases, MRS produces fewer out-liers with high simple regret than ES. We provide empirical results both for a synthetic single-task optimization problem as well as for a simulated multi-task robotic control problem.
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Bayesian optimisation has gained great popularity as a tool for optimising the parameters of machine learning algorithms and models. Somewhat ironically, setting up the hyper-parameters of Bayesian optimisation methods is notoriously hard. While reasonable practical solutions have been advanced, they can often fail to find the best optima. Surprisingly, there is little theoretical analysis of this crucial problem in the literature. To address this, we derive a cumulative regret bound for Bayesian optimisation with Gaussian processes and unknown kernel hyper-parameters in the stochastic setting. The bound, which applies to the expected improvement acquisition function and sub-Gaussian observation noise, provides us with guidelines on how to design hyper-parameter estimation methods. A simple simulation demonstrates the importance of following these guidelines.
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Many applications require optimizing an unknown, noisy function that is expensive to evaluate. We formalize this task as a multiarmed bandit problem, where the payoff function is either sampled from a Gaussian process (GP) or has low norm in a reproducing kernel Hilbert space. We resolve the important open problem of deriving regret bounds for this setting, which imply novel convergence rates for GP optimization. We analyze an intuitive Gaussian process upper confidence bound (-algorithm , and bound its cumulative regret in terms of maximal information gain, establishing a novel connection between GP optimization and experimental design. Moreover, by bounding the latter in terms of operator spectra, we obtain explicit sublinear regret bounds for many commonly used covariance functions. In some important cases, our bounds have surprisingly weak dependence on the dimensionality. In our experiments on real sensor data,-compares favorably with other heuristical GP optimization approaches.
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Bandit methods for black-box optimisation, such as Bayesian optimisation, are used in a variety of applications including hyper-parameter tuning and experiment design. Recently, multi-fidelity methods have garnered considerable attention since function evaluations have become increasingly expensive in such applications. Multi-fidelity methods use cheap approximations to the function of interest to speed up the overall opti-misation process. However, most multi-fidelity methods assume only a finite number of approximations. In many practical applications however, a continuous spectrum of approximations might be available. For instance, when tuning an expensive neural network, one might choose to approximate the cross validation performance using less data N and/or few training iterations T. Here, the approximations are best viewed as arising out of a continuous two dimensional space (N, T). In this work, we develop a Bayesian optimisa-tion method, BOCA, for this setting. We char-acterise its theoretical properties and show that it achieves better regret than than strategies which ignore the approximations. BOCA outperforms several other baselines in synthetic and real experiments .
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In this paper, we consider the challenge of maximizing an unknown function f for which evaluations are noisy and are acquired with high cost. An iterative procedure uses the previous measures to actively select the next estimation of f which is predicted to be the most useful. We focus on the case where the function can be evaluated in parallel with batches of fixed size and analyze the benefit compared to the purely sequential procedure in terms of cumulative regret. We introduce the Gaussian Process Upper Confidence Bound and Pure Exploration algorithm (GP-UCB-PE) which combines the UCB strategy and Pure Exploration in the same batch of evaluations along the parallel iterations. We prove theoretical upper bounds on the regret with batches of size K for this procedure which show the improvement of the order of ? K for fixed iteration cost over purely sequential versions. Moreover, the mul-tiplicative constants involved have the property of being dimension-free. We also confirm empirically the efficiency of GP-UCB-PE on real and synthetic problems compared to state-of-the-art competitors.
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We present a new algorithm, truncated variance reduction (TRUVAR), that treats Bayesian optimization (BO) and level-set estimation (LSE) with Gaussian processes in a unified fashion. The algorithm greedily shrinks a sum of truncated variances within a set of potential maximizers (BO) or unclassified points (LSE), which is updated based on confidence bounds. TRUVAR is effective in several important settings that are typically non-trivial to incorporate into myopic algorithms , including pointwise costs and heteroscedastic noise. We provide a general theoretical guarantee for TRUVAR covering these aspects, and use it to recover and strengthen existing results on BO and LSE. Moreover, we provide a new result for a setting where one can select from a number of noise levels having associated costs. We demonstrate the effectiveness of the algorithm on both synthetic and real-world data sets.
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Designing gaits and corresponding control policies is a key challenge in robot locomotion. Even with a viable controller parameterization, finding near-optimal parameters can be daunting. Typically, this kind of parameter optimization requires specific expert knowledge and extensive robot experiments. Automatic black-box gait optimization methods greatly reduce the need for human expertise and time-consuming design processes. Many different approaches for automatic gait optimization have been suggested to date, such as grid search and evolutionary algorithms. In this article, we thoroughly discuss multiple of these optimization methods in the context of automatic gait optimization. Moreover, we extensively evaluate Bayesian optimization, a model-based approach to black-box optimization under uncertainty, on both simulated problems and real robots. This evaluation demonstrates that Bayesian optimization is particularly suited for robotic applications, where it is crucial to find a good set of gait parameters in a small number of experiments.
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强制执行安全是许多问题的一个关键方面,这些问题涉及在不确定条件下进行顺序决策,这要求在everystep做出的决策既可以提供最佳决策信息,也可以安全。例如,我们重视药物治疗的疗效和舒适度,以及机器人控制的效率和安全性。我们认为这个优化具有绝对反馈或偏好反馈主题的未知因素函数的问题存在已知的安全约束。我们开发了一种有效的安全贝叶斯优化算法StageOpt,它将安全区域扩展和效用函数最大化分为两个不同的阶段。与在扩展和优化之间交错的现有方法相比,我们表明StageOpt更有效,并且自然地适用于更广泛的问题类。我们为满足安全约束以及收敛到最佳效用值提供理论保证。我们在各种合成实验以及临床实践中评估StageOpt。我们证明StageOpt比现有的安全优化方法更有效,并且能够在我们的临床实验中安全有效地优化脊髓刺激治疗。
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Bayesian optimization techniques have been successfully applied to robotics,planning, sensor placement, recommendation, advertising, intelligent userinterfaces and automatic algorithm configuration. Despite these successes, theapproach is restricted to problems of moderate dimension, and several workshopson Bayesian optimization have identified its scaling to high-dimensions as oneof the holy grails of the field. In this paper, we introduce a novel randomembedding idea to attack this problem. The resulting Random EMbedding BayesianOptimization (REMBO) algorithm is very simple, has important invarianceproperties, and applies to domains with both categorical and continuousvariables. We present a thorough theoretical analysis of REMBO. Empiricalresults confirm that REMBO can effectively solve problems with billions ofdimensions, provided the intrinsic dimensionality is low. They also show thatREMBO achieves state-of-the-art performance in optimizing the 47 discreteparameters of a popular mixed integer linear programming solver.
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