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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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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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 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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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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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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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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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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我们提出了一种自适应方法来构建贝叶斯推理的高斯过程,并使用昂贵的评估正演模型。我们的方法依赖于完全贝叶斯方法来训练高斯过程模型,并利用贝叶斯全局优化的预期改进思想。我们通过最大化高斯过程模型与噪声观测数据拟合的预期改进来自适应地构建训练设计。对合成数据模型问题的数值实验证明了所获得的自适应设计与固定非自适应设计相比,在前向模型推断成本的精确后验估计方面的有效性。
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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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Bayesian optimization has recently been proposed as a framework for automatically tuning the hyperparameters of machine learning models and has been shown to yield state-of-the-art performance with impressive ease and efficiency. In this paper, we explore whether it is possible to transfer the knowledge gained from previous optimizations to new tasks in order to find optimal hyperparameter settings more efficiently. Our approach is based on extending multi-task Gaussian processes to the framework of Bayesian optimization. We show that this method significantly speeds up the optimization process when compared to the standard single-task approach. We further propose a straightforward extension of our algorithm in order to jointly minimize the average error across multiple tasks and demonstrate how this can be used to greatly speed up k-fold cross-validation. Lastly, we propose an adaptation of a recently developed acquisition function, en-tropy search, to the cost-sensitive, multi-task setting. We demonstrate the utility of this new acquisition function by leveraging a small dataset to explore hyper-parameter settings for a large dataset. Our algorithm dynamically chooses which dataset to query in order to yield the most information per unit cost.
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本文提出了采集汤普森采样(ATS),这是一种基于随机过程采样多采集函数的思想的批量贝叶斯优化算法(BO)。我们通过采集函数对一组模型参数的依赖来定义该过程。 ATS在概念上简单,直接实现,与其他批处理BO方法不同,它可以用于并行化任何顺序采集功能。为了提高多模态任务的性能,我们表明ATS可以与现有技术结合以实现不同探索 - 利用交易,并考虑未决的功能评估。我们在各种基准函数和流行的梯度增强树算法的超参数优化上进行了实验。这些证明了我们的算法与两个最先进的批量BO方法的竞争力,以及它对经典并行Thompson采样BO的优势。
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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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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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Bayesian optimization has proven invaluable for black-box optimization of expensive functions. Its main limitation is its exponential complexity with respect to the dimensionality of the search space using typical kernels. Luckily, many objective functions can be decomposed into additive sub-problems, which can be optimized independently. We investigate how to automatically discover such (typically unknown) additive structure while simultaneously exploiting it through Bayesian optimization. We propose an efficient algorithm based on Metropolis-Hastings sampling and demonstrate its efficacy empirically on synthetic and real-world data sets. Throughout all our experiments we reliably discover hidden additive structure whenever it exists and exploit it to yield significantly faster convergence.
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Contemporary global optimization algorithms are based on local measures of utility, rather than a probability measure over location and value of the optimum. They thus attempt to collect low function values, not to learn about the optimum. The reason for the absence of probabilistic global optimizers is that the corresponding inference problem is intractable in several ways. This paper develops desiderata for probabilistic optimization algorithms, then presents a concrete algorithm which addresses each of the computational intractabilities with a sequence of approximations and explicitly addresses the decision problem of maximizing information gain from each evaluation.
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Bayesian optimization has become a successful tool for hyperparameter optimization of machine learning algorithms, such as support vector machines or deep neural networks. Despite its success , for large datasets, training and validating a single configuration often takes hours, days, or even weeks, which limits the achievable performance. To accelerate hyperparameter optimization , we propose a generative model for the validation error as a function of training set size, which is learned during the optimization process and allows exploration of preliminary configurations on small subsets, by extrapolating to the full dataset. We construct a Bayesian optimization procedure, dubbed FABOLAS, which models loss and training time as a function of dataset size and automatically trades off high information gain about the global optimum against computational cost. Experiments optimizing support vector machines and deep neural networks show that FABOLAS often finds high-quality solutions 10 to 100 times faster than other state-of-the-art Bayesian optimization methods or the recently proposed bandit strategy Hyperband.
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卡尔曼滤波器通常用于许多数据融合应用,包括导航,跟踪和同时定位和映射问题。然而,经常需要大量的时间和精力来调整各种卡尔曼滤波器模型参数,例如,过程噪声协方差,用于非白噪声的预白化过滤器模型等。用于调谐的传统优化技术可能陷入较差的局部最小值,并且用实际传感器数据实现可能是昂贵的。为了解决这些问题,开发了一种新的“黑匣子”贝叶斯优化策略,用于自动调整卡尔曼滤波器。在该方法中,性能的特征在于两个随机目标函数之一:当可用地面实况模型时的归一化估计误差平方(NEES),或者当仅有传感器数据可用时的归一化创新误差平方(NIS)。通过智能地对参数空间进行采样学习并利用NEES / NIS成本的非参数高斯过程替代函数,贝叶斯优化可以有效地识别多个局部最小值并对其结果提供不确定性量化。
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许多对科学计算和机器学习感兴趣的概率模型具有昂贵的黑盒可能性,这些可能性阻止了贝叶斯推理的标准技术的应用,例如MCMC,其需要接近梯度或大量可能性评估。我们在这里介绍一种新颖的样本有效推理框架,VariationalBayesian Monte Carlo(VBMC)。 VBMC将变分推理与基于高斯过程的有源采样贝叶斯积分结合起来,使用latterto有效逼近变分目标中的难以求的积分。我们的方法产生了后验分布的非参数近似和模型证据的近似下界,对模型选择很有用。我们在几种合成可能性和神经元模型上展示VBMC,其中包含来自真实神经元的数据。在所有测试的问题和维度(高达$ D = 10 $)中,VBMC始终如一地通过有限的可能性评估预算重建后验证和模型证据,而不像其他仅在非常低维度下工作的方法。我们的框架作为一种新颖的工具,具有昂贵的黑盒可能性,可用于后期模型推理。
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当只能获得有限数量的noisylog-likelihood评估时,我们考虑贝叶斯推断。例如,当基于复杂模拟器的统计模型适合于数据时,这发生,并且使用合成似然(SL)来使用计算成本高的前向模拟来形成噪声对数似然估计。我们将推理任务构建为贝叶斯序列设计问题,其中对数似然函数使用分层高斯过程(GP)代理模型进行建模,该模型用于有效地选择其他对数似然评估位置。最近在批处理贝叶斯优化中取得了进展,我们开发了各种顺序策略,其中自适应地选择多个模拟以最小化预期或中值损失函数,从而测量所得到的后验中的不确定性。我们从理论上和经验上分析了所得方法的性质。玩具问题和三个模拟模型的实验表明我们的方法是稳健的,高度可并行的,并且样本有效。
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