Entropy Search (ES) and Predictive Entropy Search (PES) are popular and empirically successful Bayesian Optimization techniques. Both rely on a compelling information-theoretic motivation , and maximize the information gained about the arg max of the unknown function; yet, both are plagued by the expensive computation for estimating entropies. We propose a new criterion , Max-value Entropy Search (MES), that instead uses the information about the maximum function value. We show relations of MES to other Bayesian optimization methods, and establish a regret bound. We observe that MES maintains or improves the good empirical performance of ES/PES, while tremendously lightening the computational burden. In particular, MES is much more robust to the number of samples used for computing the entropy, and hence more efficient for higher dimensional problems.
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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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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 posterior estimate of the objective. However, there are several different parameterized acquisition functions in the literature, and it is often unclear which one to use. Instead of using a single acquisition function, we adopt a portfolio of acquisition functions governed by an online multi-armed bandit strategy. We propose several portfolio strategies, the best of which we call GP-Hedge, and show that this method outperforms the best individual acquisition function. We also provide a theoretical bound on the algorithm's performance .
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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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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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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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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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在本文中,我们考虑了高斯过程(GP)优化的问题,增加了鲁棒性要求:返回点可能受到厌恶扰动,并且我们要求函数值即使在此扰动之后仍保持尽可能高。该问题的动机是在优化和实施阶段期间的基础功能不同,或者当人们有兴趣找到优于单个点的良好输入的整个区域时。我们证明标准GP优化算法没有表现出所需的鲁棒性,并为此目的提供了基于置信限制的算法StableOpt。大力建立StableOpt所需的样本数量以找到最佳点,我们用独立于算法的下限来补充这种保证。我们使用真实世界的数据集实验性地展示了几个感兴趣的潜在应用程序,并且我们展示了SattableOpt一直在寻找一个稳定的最大化器,其中几个baseline方法失败。
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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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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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This paper explores the problem of path planning under uncertainty. Specifically, we consider online receding horizon based planners that need to operate in a latent environment where the latent information can be modeled via Gaussian Processes. Online path planning in latent environments is challenging since the robot needs to explore the environment to get a more accurate model of latent information for better planning later and also achieves the task as quick as possible. We propose UCB style algorithms that are popular in the bandit settings and show how those analyses can be adapted to the online robotic path planning problems. The proposed algorithm trades-off exploration and exploitation in near-optimal manner and has appealing no-regret properties. We demonstrate the efficacy of the framework on the application of aircraft flight path planning when the winds are partially observed.
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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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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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机器学习算法的性能主要取决于识别一组好的超参数。虽然最近的方法使用贝叶斯优化来自适应地选择配置,但我们专注于通过自适应资源分配和早期停止来加速随机搜索。我们将超参数优化表示为纯探索非随机无限制武装强盗问题,其中将迭代,数据样本或特征等预定义资源分配给随机采样配置。我们为该框架引入了一种新的算法Hyperband,并对其理论属性进行了分析,提供了几种理想的保证。此外,我们在一系列超参数优化问题上将Hyperband与流行的贝叶斯优化方法进行比较。我们观察到Hyperband可以提供超过我们竞争对手的各种深度学习和基于内核的学习问题的数量级加速。
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AutoML已成为当今大多数领先的云服务提供商提供的热门服务。在本文中,我们从\ emph {服务提供商的角度}关注AutoML问题,主要得益于以下实际考虑:当AutoML服务需要同时为{\ em多个设备}提供{\ em多个用户}时,我们如何以有效的方式分配这些设备?我们专注于GP-EI,这是自动模型选择和超参数调整的最常用算法之一,由Google Vizer等系统使用。本文的技术贡献是GP-EI的第一个多设备,多租户算法,它知道\ emph {多}计算设备和多个用户共享同一组计算设备。从理论上讲,给定$ N $用户和$ M $设备,我们获得了$ O的后悔限制((\ text {\ bf {MIU}}(T,K)+ M)\ frac {N ^ 2} {M}) $,其中$ \ text {\ bf {MIU}}(T,K)$是指协方差矩阵$ K $的最大增量不确定性上升$ T $。根据经验,我们评估了我们的算法两种自动模型选择的应用,并表明我们的算法显着优于独立服务用户的策略。此外,当有多个计算设备可用时,当用户数远大于用户数时,我们实现了近线性加速。设备数量。
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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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许多现实世界的应用可以被构造为多目标优化问题,我们希望同时针对多个标准进行优化。当所讨论的功能的评估是昂贵的时,用于多目标设置的贝叶斯优化技术是相关的。用于多目标优化的传统方法,无论是贝叶斯还是其他方式,都旨在恢复这些目标的帕累托前沿。然而,在某些情况下,由于外部考虑,从业者可能希望仅在帕累托前沿的特定区域中识别帕累托最优点。在这项工作中,我们提出了一种策略,该策略基于解决该问题的目标的随机标量化。虽然在计算上与其他方法相似或相似,但我们的方法足够灵活,可以从帕累托前沿或整个前端的特定子集中进行采样。我们还在多目标背景下引入了一种遗憾的新观念,表明我们的策略存在次线性遗憾。我们尝试了合成和现实问题,并展示了我们提出的算法的灵活性,可扩展性和遗憾的优越性能。
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