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Gaussian Process Optimization in the Bandit Setting: No Regret and Experimental Design
"... 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 RKHS norm. We resolve the important open problem of deriving regre ..."
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Cited by 118 (11 self)
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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 RKHS norm. We resolve the important open problem of deriving regret bounds for this setting, which imply novel convergence rates for GP optimization. We analyze GPUCB, an intuitive upperconfidence based 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, GPUCB compares favorably with other heuristical GP optimization approaches. 1.
Adaptive submodularity: Theory and applications in active learning and stochastic optimization
 J. Artificial Intelligence Research
, 2011
"... Many problems in artificial intelligence require adaptively making a sequence of decisions with uncertain outcomes under partial observability. Solving such stochastic optimization problems is a fundamental but notoriously difficult challenge. In this paper, we introduce the concept of adaptive subm ..."
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Cited by 64 (15 self)
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Many problems in artificial intelligence require adaptively making a sequence of decisions with uncertain outcomes under partial observability. Solving such stochastic optimization problems is a fundamental but notoriously difficult challenge. In this paper, we introduce the concept of adaptive submodularity, generalizing submodular set functions to adaptive policies. We prove that if a problem satisfies this property, a simple adaptive greedy algorithm is guaranteed to be competitive with the optimal policy. In addition to providing performance guarantees for both stochastic maximization and coverage, adaptive submodularity can be exploited to drastically speed up the greedy algorithm by using lazy evaluations. We illustrate the usefulness of the concept by giving several examples of adaptive submodular objectives arising in diverse AI applications including management of sensing resources, viral marketing and active learning. Proving adaptive submodularity for these problems allows us to recover existing results in these applications as special cases, improve approximation guarantees and handle natural generalizations. 1.
AutoWEKA: Combined Selection and Hyperparameter Optimization of Classification Algorithms
"... Many different machine learning algorithms exist; taking into account each algorithm’s hyperparameters, there is a staggeringly large number of possible alternatives overall. We consider the problem of simultaneously selecting a learning algorithm and setting its hyperparameters, going beyond previo ..."
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Cited by 27 (8 self)
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Many different machine learning algorithms exist; taking into account each algorithm’s hyperparameters, there is a staggeringly large number of possible alternatives overall. We consider the problem of simultaneously selecting a learning algorithm and setting its hyperparameters, going beyond previous work that attacks these issues separately. We show that this problem can be addressed by a fully automated approach, leveraging recent innovations in Bayesian optimization. Specifically, we consider a wide range of feature selection techniques (combining 3 search and 8 evaluator methods) and all classification approaches implemented in WEKA’s standard distribution, spanning 2 ensemble methods, 10 metamethods, 27 base classifiers, and hyperparameter settings for each classifier. On each of 21 popular datasets from the UCI repository, the KDD Cup 09, variants of the MNIST dataset and CIFAR10, we show classification performance often much better than using standard selection and hyperparameter optimization methods. We hope that our approach will help nonexpert users to more effectively identify machine learning algorithms and hyperparameter settings appropriate to their applications, and hence to achieve improved performance.
Portfolio Allocation for Bayesian Optimization
"... 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 blackbox optimization scenarios. It uses Bayesian methods t ..."
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Cited by 24 (14 self)
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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 blackbox 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 multiarmed bandit strategy. We propose several portfolio strategies, the best of which we call GPHedge, and show that this method outperforms the best individual acquisition function. We also provide a theoretical bound on the algorithm’s performance. 1
Informationtheoretic regret bounds for Gaussian process optimization in the bandit setting. Information Theory
 IEEE Transactions on
, 2012
"... Abstract—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 th ..."
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Cited by 23 (2 self)
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Abstract—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. Index Terms—Bandit problems, Bayesian prediction, experimental design, Gaussian process (GP), information gain,
Exponential regret bounds for Gaussian process bandits with deterministic observations
 In ICML
, 2012
"... This paper analyzes the problem of Gaussian process (GP) bandits with deterministic observations. The analysis uses a branch and bound algorithm that is related to the UCB algorithm of (Srinivas et al., 2010). For GPs with Gaussian observation noise, with variance strictly greater than zero, (Sriniv ..."
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Cited by 16 (9 self)
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This paper analyzes the problem of Gaussian process (GP) bandits with deterministic observations. The analysis uses a branch and bound algorithm that is related to the UCB algorithm of (Srinivas et al., 2010). For GPs with Gaussian observation noise, with variance strictly greater than zero, (Srinivas et al., 2010) proved that the regret ( ) vanishes at the approximate 1√t rate of O, where t is the number of observations. To complement their result, we attack the deterministic case and attain a much faster exponential convergence rate. Under some regularity assumptions, we show that the regret decreases asymptotically according to O e − τt (ln t) d/4 with high probability. Here, d is the dimension of the search space and τ is a constant that depends on the behaviour of the objective function near its global maximum. 1.
Contextual Gaussian Process Bandit Optimization
"... How should we design experiments to maximize performance of a complex system, taking into account uncontrollable environmental conditions? How should we select relevant documents (ads) to display, given information about the user? These tasks can be formalized as contextual bandit problems, where at ..."
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Cited by 15 (2 self)
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How should we design experiments to maximize performance of a complex system, taking into account uncontrollable environmental conditions? How should we select relevant documents (ads) to display, given information about the user? These tasks can be formalized as contextual bandit problems, where at each round, we receive context (about the experimental conditions, the query), and have to choose an action (parameters, documents). The key challenge is to trade off exploration by gathering data for estimating the mean payoff function over the contextaction space, and to exploit by choosing an action deemed optimal based on the gathered data. We model the payoff function as a sample from a Gaussian process defined over the joint contextaction space, and develop CGPUCB, an intuitive upperconfidence style algorithm. We show that by mixing and matching kernels for contexts and actions, CGPUCB can handle a variety of practical applications. We further provide generic tools for deriving regret bounds when using such composite kernel functions. Lastly, we evaluate our algorithm on two case studies, in the context of automated vaccine design and sensor management. We show that contextsensitive optimization outperforms no or naive use of context. 1
Batch Bayesian Optimization via Simulation Matching
"... Bayesian optimization methods are often used to optimize unknown functions that are costly to evaluate. Typically, these methods sequentially select inputs to be evaluated one at a time based on a posterior over the unknown function that is updated after each evaluation. In many applications, howeve ..."
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Cited by 14 (6 self)
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Bayesian optimization methods are often used to optimize unknown functions that are costly to evaluate. Typically, these methods sequentially select inputs to be evaluated one at a time based on a posterior over the unknown function that is updated after each evaluation. In many applications, however, it is desirable to perform multiple evaluations in parallel, which requires selecting batches of multiple inputs to evaluate at once. In this paper, we propose a novel approach to batch Bayesian optimization, providing a policy for selecting batches of inputs with the goal of optimizing the function as efficiently as possible. The key idea is to exploit the availability of highquality and efficient sequential policies, by using MonteCarlo simulation to select input batches that closely match their expected behavior. Our experimental results on six benchmarks show that the proposed approach significantly outperforms two baselines and can lead to large advantages over a top sequential approach in terms of performance per unit time. 1
Bayesian Optimization in High Dimensions via Random Embeddings
"... Bayesian optimization techniques have been successfully applied to robotics, planning, sensor placement, recommendation, advertising, intelligent user interfaces and automatic algorithm configuration. Despite these successes, the approach is restricted to problems of moderate dimension, and several ..."
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Cited by 11 (6 self)
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Bayesian optimization techniques have been successfully applied to robotics, planning, sensor placement, recommendation, advertising, intelligent user interfaces and automatic algorithm configuration. Despite these successes, the approach is restricted to problems of moderate dimension, and several workshops on Bayesian optimization have identified its scaling to high dimensions as one of the holy grails of the field. In this paper, we introduce a novel random embedding idea to attack this problem. The resulting Random EMbedding Bayesian Optimization (REMBO) algorithm is very simple and applies to domains with both categorical and continuous variables. The experiments demonstrate that REMBO can effectively solve highdimensional problems, including automatic parameter configuration of a popular mixed integer linear programming solver.
Adaptive Hamiltonian and Riemann Manifold Monte Carlo Samplers
"... In this paper we address the widelyexperienced difficulty in tuning Monte Carlo sampler based on simulating Hamiltonian dynamics. We develop an algorithm that allows for the adaptation of Hamiltonian and Riemann manifold Hamiltonian Monte Carlo samplers using Bayesian optimization that allows for in ..."
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Cited by 11 (4 self)
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In this paper we address the widelyexperienced difficulty in tuning Monte Carlo sampler based on simulating Hamiltonian dynamics. We develop an algorithm that allows for the adaptation of Hamiltonian and Riemann manifold Hamiltonian Monte Carlo samplers using Bayesian optimization that allows for infinite adaptation of the parameters of these samplers. We show that the resulting samplers are ergodic, and that the use of our adaptive algorithms makes it easy to obtain more efficient samplers, in some cases precluding the need for more complex solutions. Hamiltonianbased Monte Carlo samplers are widely known to be an excellent choice of MCMC method, and we aim with this paper to remove a key obstacle towards the more widespread use of these samplers in practice. 1.