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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We propose a novel information-theoretic approach for Bayesian optimization called Predictive Entropy Search (PES). At each iteration, PES selects the next evaluation point that maximizes the expected information gained with respect to the global maximum. PES codifies this intractable acquisition function in terms of the expected reduction in the differential entropy of the predictive distribution. This reformulation allows PES to obtain approximations that are both more accurate and efficient than other alternatives such as Entropy Search (ES). Furthermore , PES can easily perform a fully Bayesian treatment of the model hy-perparameters while ES cannot. We evaluate PES in both synthetic and real-world applications, including optimization problems in machine learning, finance, biotechnology, and robotics. We show that the increased accuracy of PES leads to significant gains in optimization performance.
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随机实验是评估变化对现实世界系统影响的黄金标准。这些测试中的数据可能难以收集,结果可能具有高度差异,从而导致潜在的大量测量误差。贝叶斯优化是一种有效优化多个连续参数的有前途的技术,但是当噪声水平高时,现有方法降低了性能,限制了其对多个随机实验的适用性。我们得到了一个表达式,用于预期的改进,具有噪声观察和噪声约束的批量优化,并开发了一种准蒙特卡罗近似,可以有效地进行优化。使用合成函数进行的仿真表明,噪声约束问题的优化性能优于现有方法。我们通过在Facebook上进行的两个真实的实验来进一步证明该方法的有效性:优化排名系统和优化服务器编译器标志。
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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 knowledge, PPES is the first non-greedy batch Bayesian optimization strategy. We demonstrate the benefit of this approach in optimization performance on both synthetic and real world applications , including problems in machine learning, rocket science and robotics.
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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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贝叶斯优化是一种优化目标函数的方法,需要花费很长时间(几分钟或几小时)来评估。它最适合于在小于20维的连续域上进行优化,并且在功能评估中容忍随机噪声。它构建了目标的替代品,并使用贝叶斯机器学习技术,高斯过程回归量化该替代品中的不确定性,然后使用从该代理定义的获取函数来决定在何处进行抽样。在本教程中,我们描述了贝叶斯优化的工作原理,包括高斯过程回归和三种常见的采集功能:预期改进,熵搜索和知识梯度。然后,我们讨论了更先进的技术,包括在并行,多保真和多信息源优化,昂贵的评估约束,随机环境条件,多任务贝叶斯优化以及包含衍生信息的情况下运行多功能评估。最后,我们讨论了贝叶斯优化软件和该领域未来的研究方向。在我们的教程材料中,我们提供了对噪声评估的预期改进的时间化,超出了无噪声设置,在更常用的情况下。这种概括通过正式的决策理论论证来证明,与先前的临时修改形成鲜明对比。
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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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Recent work on Bayesian optimization has shown its effectiveness in global optimization of difficult black-box objective functions. Many real-world optimization problems of interest also have constraints which are unknown a priori. In this paper, we study Bayesian optimization for constrained problems in the general case that noise may be present in the constraint functions, and the objective and constraints may be evaluated independently. We provide motivating practical examples, and present a general framework to solve such problems. We demonstrate the effectiveness of our approach on optimizing the performance of online latent Dirichlet allocation subject to topic sparsity constraints, tuning a neural network given test-time memory constraints, and optimizing Hamiltonian Monte Carlo to achieve maximal effectiveness in a fixed time, subject to passing standard convergence diagnostics.
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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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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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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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贝叶斯决策理论概述了一个严格的框架,用于在最大化模型后验的预期效用的基础上进行最优决策。然而,从业者通常无法获得完整的后验和诉诸于近似的推理策略。在这种情况下,在执行推断时考虑最终决策制定任务允许校准后验近似以最大化效用。我们提出了一个自动化管道,它将连续的实用程序与变分推理算法相结合,以便考虑决策。我们提供了实用的策略来实现并最大化增益,并在校准特定实用程序的近似值时凭经验证明了一致性。
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我们考虑学习嘈杂的黑盒功能超过给定阈值的水平集的问题。为了有效地重建水平集,我们研究了高斯过程(GP)元模型。我们的重点是强随机采样器,特别是重尾模拟噪声和低信噪比。为了防止噪声错误指定,我们评估了三个变量的性能:(i)具有Student-$ t $观察值的GP; (ii)学生 - $ t $流程(TP); (iii)分类GP对响应的符号进行建模。作为第四个扩展,我们研究具有单调性约束的GP代理,这些约束在已知连接的级别集时是相关的。结合这些模型,我们分析了几个采集函数,用于指导顺序实验设计,将现有的逐步不确定性减少标准扩展到随机轮廓发现环境。这也促使我们开发(近似)更新公式以有效地计算取代函数。我们的方案通过在1-6维度中使用各种合成实验进行基准测试。我们还考虑应用水平集估计来确定最佳的运动政策和百慕大金融期权的估值。
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When monitoring spatial phenomena, which can often be modeled as Gaussian processes (GPs), choosing sensor locations is a fundamental task. There are several common strategies to address this task, for example, geometry or disk models, placing sensors at the points of highest entropy (vari-ance) in the GP model, and AD D-, or E-optimal design. In this paper, we tackle the combinatorial optimization problem of maximizing the mutual information between the chosen locations and the locations which are not selected. We prove that the problem of finding the configuration that maximizes mutual information is NP-complete. To address this issue, we describe a polynomial-time approximation that is within (1 − 1/e) of the optimum by exploiting the submodularity of mutual information. We also show how submodularity can be used to obtain online bounds, and design branch and bound search procedures. We then extend our algorithm to exploit lazy evaluations and local structure in the GP, yielding significant speedups. We also extend our approach to find placements which are robust against node failures and uncertainties in the model. These extensions are again associated with rigorous theoretical approximation guarantees, exploiting the submodu-larity of the objective function. We demonstrate the advantages of our approach towards optimizing mutual information in a very extensive empirical study on two real-world data sets.
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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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Bayesian optimization has been successful at global optimization of expensive-to-evaluate multimodal objective functions. However, unlike most optimization methods, Bayesian optimization typically does not use derivative information. In this paper we show how Bayesian optimization can exploit derivative information to find good solutions with fewer objective function evaluations. In particular, we develop a novel Bayesian optimization algorithm, the derivative-enabled knowledge-gradient (d-KG), which is one-step Bayes-optimal, asymptotically consistent, and provides greater one-step value of information than in the derivative-free setting. d-KG accommodates noisy and incomplete derivative information, comes in both sequential and batch forms, and can optionally reduce the computational cost of inference through automatically selected retention of a single directional derivative. We also compute the d-KG acquisition function and its gradient using a novel fast discretization-free technique. We show d-KG provides state-of-the-art performance compared to a wide range of optimization procedures with and without gradients, on benchmarks including logistic regression, deep learning, kernel learning, and k-nearest neighbors.
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贝叶斯优化(BO)是用于解决挑战优化任务的流行算法。它适用于目标函数评估成本昂贵的问题,可能无法以精确的形式提供,没有梯度信息并且可能返回噪声值。不同版本的算法在采集功能的选择上有所不同,它建议点在下一步查询目标。最初,研究人员专注于基于改进的收购,而最近注意力已经转移了计算上昂贵的信息理论措施。在本文中,我们提出了两个主要的文献贡献。首先,我们提出了一种新的基于改进的采集功能,该功能可以推荐高可信度提高的查询点。所提出的算法在全局优化文献的大量基准函数上进行评估,其中至少与当前最先进的采集函数一样,并且通常更好。这表明它是BO的强大默认选择。然后将新颖的策略与有用的全局优化求解器进行比较,以确认BO方法通过保持数量的函数评估较小来降低优化的计算成本。第二个主要贡献代表了对精准医学的应用,其中兴趣在于估计人肺血循环系统的偏微分方程模型的参数。一旦推断,这些参数可以帮助临床医生诊断患有肺动脉高压的患者,而无需通过右心导管插入术的标准侵入性程序,这可导致toside效应和并发症(例如严重疼痛,内出血,血栓形成)。
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近似贝叶斯计算(ABC)是贝叶斯推理的一种方法,当可能性不可用时,但是可以从模型中进行模拟。然而,许多ABC算法需要大量的模拟,这可能是昂贵的。为了降低计算成本,已经提出了贝叶斯优化(BO)和诸如高斯过程的模拟模型。贝叶斯优化使人们可以智能地决定在哪里评估模型下一个,但是常见的BO策略不是为了估计后验分布而设计的。我们的论文解决了文献中的这一差距。我们建议计算ABC后验密度的不确定性,这是因为缺乏模拟来准确估计这个数量,并且定义了测量这种不确定性的aloss函数。然后,我们建议选择下一个评估位置,以尽量减少预期的损失。实验表明,与普通BO策略相比,所提出的方法通常产生最准确的近似。
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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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概率二分算法基于从嘈杂的oracle响应中获得的知识来执行根查找。我们考虑广义PBA设置(G-PBA),其中oracle的统计分布是未知的和位置依赖的,因此模型推理和贝叶斯知识更新必须同时进行。为此,我们建议通过构建基础逻辑回归步骤的统计代理来利用典型oracle的空间结构。我们研究了几个非参数代理,包括二项式高斯过程(B-GP),多项式,核和样条Logistic回归。与此同时,我们开发了自适应平衡学习oracle分布和学习根的策略。我们的一个建议模仿了B-GP的主动学习,并提供了一种新颖的前瞻预测方差公式。我们用空间PBA算法得到的相对于早期G-PBA模型的增益用合成的例子和来自贝尔丹期权定价的具有挑战性的随机根发现问题来说明。
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