arXiv subjects

Peter I. Frazier

Publications and source records attributed to Peter I. Frazier.

50 records · Page 3Linked to original sources

arXiv2016

The Bayesian Linear Information Filtering Problem

We present a Bayesian sequential decision-making formulation of the information filtering problem, in which an algorithm presents items (news articles, scientific papers, tweets) arriving in a stream, and learns relevance from user feedback on presented items. We model user preferences using a Bayesian linear model, similar in spirit to a Bayesian linear bandit. We compute a computational upper bound on the value of the optimal policy, which allows computing an optimality gap for implementable policies. We then use this analysis as motivation in introducing a pair of new Decompose-Then-Decide (DTD) heuristic policies, DTD-Dynamic-Programming (DTD-DP) and DTD-Upper-Confidence-Bound (DTD-UCB). We compare DTD-DP and DTD-UCB against several benchmarks on real and simulated data, demonstrating significant improvement, and show that the achieved performance is close to the upper bound.

cs.LG↗

arXiv2016

Warm Starting Bayesian Optimization

We develop a framework for warm-starting Bayesian optimization, that reduces the solution time required to solve an optimization problem that is one in a sequence of related problems. This is useful when optimizing the output of a stochastic simulator that fails to provide derivative information, for which Bayesian optimization methods are well-suited. Solving sequences of related optimization problems arises when making several business decisions using one optimization model and input data collected over different time periods or markets. While many gradient-based methods can be warm started by initiating optimization at the solution to the previous problem, this warm start approach does not apply to Bayesian optimization methods, which carry a full metamodel of the objective function from iteration to iteration. Our approach builds a joint statistical model of the entire collection of related objective functions, and uses a value of information calculation to recommend points to evaluate.

stat.ML↗

arXiv2016

Multi-Step Bayesian Optimization for One-Dimensional Feasibility Determination

Bayesian optimization methods allocate limited sampling budgets to maximize expensive-to-evaluate functions. One-step-lookahead policies are often used, but computing optimal multi-step-lookahead policies remains a challenge. We consider a specialized Bayesian optimization problem: finding the superlevel set of an expensive one-dimensional function, with a Markov process prior. We compute the Bayes-optimal sampling policy efficiently, and characterize the suboptimality of one-step lookahead. Our numerical experiments demonstrate that the one-step lookahead policy is close to optimal in this problem, performing within 98% of optimal in the experimental settings considered.

math.OC↗

arXiv2016

Stratified Bayesian Optimization

We consider derivative-free black-box global optimization of expensive noisy functions, when most of the randomness in the objective is produced by a few influential scalar random inputs. We present a new Bayesian global optimization algorithm, called Stratified Bayesian Optimization (SBO), which uses this strong dependence to improve performance. Our algorithm is similar in spirit to stratification, a technique from simulation, which uses strong dependence on a categorical representation of the random input to reduce variance. We demonstrate in numerical experiments that SBO outperforms state-of-the-art Bayesian optimization benchmarks that do not leverage this dependence.

cs.LG↗

arXiv2015

Bayes-Optimal Effort Allocation in Crowdsourcing: Bounds and Index Policies

We consider effort allocation in crowdsourcing, where we wish to assign labeling tasks to imperfect homogeneous crowd workers to maximize overall accuracy in a continuous-time Bayesian setting, subject to budget and time constraints. The Bayes-optimal policy for this problem is the solution to a partially observable Markov decision process, but the curse of dimensionality renders the computation infeasible. Based on the Lagrangian Relaxation technique in Adelman & Mersereau (2008), we provide a computationally tractable instance-specific upper bound on the value of this Bayes-optimal policy, which can in turn be used to bound the optimality gap of any other sub-optimal policy. In an approach similar in spirit to the Whittle index for restless multiarmed bandits, we provide an index policy for effort allocation in crowdsourcing and demonstrate numerically that it outperforms other stateof- arts and performs close to optimal solution.

cs.LG↗

arXiv2015

Probabilistic Group Testing under Sum Observations: A Parallelizable 2-Approximation for Entropy Loss

We consider the problem of group testing with sum observations and noiseless answers, in which we aim to locate multiple objects by querying the number of objects in each of a sequence of chosen sets. We study a probabilistic setting with entropy loss, in which we assume a joint Bayesian prior density on the locations of the objects and seek to choose the sets queried to minimize the expected entropy of the Bayesian posterior distribution after a fixed number of questions. We present a new non-adaptive policy, called the dyadic policy, show it is optimal among non-adaptive policies, and is within a factor of two of optimal among adaptive policies. This policy is quick to compute, its nonadaptive nature makes it easy to parallelize, and our bounds show it performs well even when compared with adaptive policies. We also study an adaptive greedy policy, which maximizes the one-step expected reduction in entropy, and show that it performs at least as well as the dyadic policy, offering greater query efficiency but reduced parallelism. Numerical experiments demonstrate that both procedures outperform a divide-and-conquer benchmark policy from the literature, called sequential bifurcation, and show how these procedures may be applied in a stylized computer vision problem.

cs.IT↗

arXiv2015

Asymptotic Validity of the Bayes-Inspired Indifference Zone Procedure: The Non-Normal Known Variance Case

We consider the indifference-zone (IZ) formulation of the ranking and selection problem in which the goal is to choose an alternative with the largest mean with guaranteed probability, as long as the difference between this mean and the second largest exceeds a threshold. Conservatism leads classical IZ procedures to take too many samples in problems with many alternatives. The Bayes-inspired Indifference Zone (BIZ) procedure, proposed in Frazier (2014), is less conservative than previous procedures, but its proof of validity requires strong assumptions, specifically that samples are normal, and variances are known with an integer multiple structure. In this paper, we show asymptotic validity of a slight modification of the original BIZ procedure as the difference between the best alternative and the second best goes to zero,when the variances are known and finite, and samples are independent and identically distributed, but not necessarily normal.

stat.ME↗

arXiv2015

Bayesian optimization for materials design

We introduce Bayesian optimization, a technique developed for optimizing time-consuming engineering simulations and for fitting machine learning models on large datasets. Bayesian optimization guides the choice of experiments during materials design and discovery to find good material designs in as few experiments as possible. We focus on the case when materials designs are parameterized by a low-dimensional vector. Bayesian optimization is built on a statistical technique called Gaussian process regression, which allows predicting the performance of a new design based on previously tested designs. After providing a detailed introduction to Gaussian process regression, we introduce two Bayesian optimization methods: expected improvement, for design problems with noise-free evaluations; and the knowledge-gradient method, which generalizes expected improvement and may be used in design problems with noisy evaluations. Both methods are derived using a value-of-information analysis, and enjoy one-step Bayes-optimality.

stat.ML↗

arXiv2015

Clustering via Content-Augmented Stochastic Blockmodels

Much of the data being created on the web contains interactions between users and items. Stochastic blockmodels, and other methods for community detection and clustering of bipartite graphs, can infer latent user communities and latent item clusters from this interaction data. These methods, however, typically ignore the items' contents and the information they provide about item clusters, despite the tendency of items in the same latent cluster to share commonalities in content. We introduce content-augmented stochastic blockmodels (CASB), which use item content together with user-item interaction data to enhance the user communities and item clusters learned. Comparisons to several state-of-the-art benchmark methods, on datasets arising from scientists interacting with scientific articles, show that content-augmented stochastic blockmodels provide highly accurate clusters with respect to metrics representative of the underlying community structure.

stat.ML↗

arXiv2015

Exploration vs. Exploitation in the Information Filtering Problem

We consider information filtering, in which we face a stream of items too voluminous to process by hand (e.g., scientific articles, blog posts, emails), and must rely on a computer system to automatically filter out irrelevant items. Such systems face the exploration vs. exploitation tradeoff, in which it may be beneficial to present an item despite a low probability of relevance, just to learn about future items with similar content. We present a Bayesian sequential decision-making model of this problem, show how it may be solved to optimality using a decomposition to a collection of two-armed bandit problems, and show structural results for the optimal policy. We show that the resulting method is especially useful when facing the cold start problem, i.e., when filtering items for new users without a long history of past interactions. We then present an application of this information filtering method to a historical dataset from the arXiv.org repository of scientific articles.

math.OC↗

arXiv2014

A Markov Decision Process Analysis of the Cold Start Problem in Bayesian Information Filtering

We consider the information filtering problem, in which we face a stream of items, and must decide which ones to forward to a user to maximize the number of relevant items shown, minus a penalty for each irrelevant item shown. Forwarding decisions are made separately in a personalized way for each user. We focus on the cold-start setting for this problem, in which we have limited historical data on the user's preferences, and must rely on feedback from forwarded articles to learn which the fraction of items relevant to the user in each of several item categories. Performing well in this setting requires trading exploration vs. exploitation, forwarding items that are likely to be irrelevant, to allow learning that will improve later performance. In a Bayesian setting, and using Markov decision processes, we show how the Bayes-optimal forwarding algorithm can be computed efficiently when the user will examine each forwarded article, and how an upper bound on the Bayes-optimal procedure and a heuristic index policy can be obtained for the setting when the user will examine only a limited number of forwarded items. We present results from simulation experiments using parameters estimated using historical data from arXiv.org.

cs.LG↗

arXiv2014

A New Optimal Stepsize For Approximate Dynamic Programming

Approximate dynamic programming (ADP) has proven itself in a wide range of applications spanning large-scale transportation problems, health care, revenue management, and energy systems. The design of effective ADP algorithms has many dimensions, but one crucial factor is the stepsize rule used to update a value function approximation. Many operations research applications are computationally intensive, and it is important to obtain good results quickly. Furthermore, the most popular stepsize formulas use tunable parameters and can produce very poor results if tuned improperly. We derive a new stepsize rule that optimizes the prediction error in order to improve the short-term performance of an ADP algorithm. With only one, relatively insensitive tunable parameter, the new rule adapts to the level of noise in the problem and produces faster convergence in numerical experiments.

math.OC↗

arXiv2012

Distance Dependent Infinite Latent Feature Models

Latent feature models are widely used to decompose data into a small number of components. Bayesian nonparametric variants of these models, which use the Indian buffet process (IBP) as a prior over latent features, allow the number of features to be determined from the data. We present a generalization of the IBP, the distance dependent Indian buffet process (dd-IBP), for modeling non-exchangeable data. It relies on distances defined between data points, biasing nearby data to share more features. The choice of distance measure allows for many kinds of dependencies, including temporal and spatial. Further, the original IBP is a special case of the dd-IBP. In this paper, we develop the dd-IBP and theoretically characterize its feature-sharing properties. We derive a Markov chain Monte Carlo sampler for a linear Gaussian model with a dd-IBP prior and study its performance on several non-exchangeable data sets.

stat.ML↗

arXiv2011

Distance Dependent Chinese Restaurant Processes

We develop the distance dependent Chinese restaurant process (CRP), a flexible class of distributions over partitions that allows for non-exchangeability. This class can be used to model many kinds of dependencies between data in infinite clustering models, including dependencies across time or space. We examine the properties of the distance dependent CRP, discuss its connections to Bayesian nonparametric mixture models, and derive a Gibbs sampler for both observed and mixture settings. We study its performance with three text corpora. We show that relaxing the assumption of exchangeability with distance dependent CRPs can provide a better fit to sequential data. We also show its alternative formulation of the traditional CRP leads to a faster-mixing Gibbs sampling algorithm than the one based on the original formulation.

stat.ML↗