Peter I. Frazier

A Knowledge-Gradient Policy for Sequential Information Collection

In a sequential Bayesian ranking and selection problem with independent normal populations and common known variance, we study a previously introduced measurement policy which we refer to as the knowledge-gradient policy. This policy myopically maximizes the expected increment in the value of information in each time period, where the value is measured according to the terminal utility function. We show that the knowledge-gradient policy is optimal both when the horizon is a single time period and in the limit as the horizon extends to infinity. We show furthermore that, in some special cases, the knowledge-gradient policy is optimal regardless of the length of any given fixed total sampling horizon. We bound the knowledge-gradient policys suboptimality in the remaining cases, and show through simulations that it performs competitively with or significantly better than other policies.

The Knowledge Gradient Algorithm for a General Class of Online Learning Problems

We derive a one-period look-ahead policy for finite-and infinite-horizon online optimal learning problems with Gaussian rewards. Our approach is able to handle the case where our prior beliefs about the rewards are correlated, which is not handled by traditional multiarmed bandit methods. Experiments show that our KG policy performs competitively against the best-known approximation to the optimal policy in the classic bandit problem, and it outperforms many learning policies in the correlated case.

Paradoxes in Learning and the Marginal Value of Information

We consider the Bayesian ranking and selection problem, in which one wishes to allocate an information collection budget as efficiently as possible to choose the best among several alternatives. In this problem, the marginal value of information is not concave, leading to algorithmic difficulties and apparent paradoxes. Among these paradoxes is that when there are many identical alternatives, it is often better to ignore some completely and focus on a smaller number than it is to spread the measurement budget equally across all the alternatives. We analyze the consequences of this nonconcavity in several classes of ranking and selection problems, showing that the value of information is “eventually concave,” i.e., concave when the number of measurements of each alternative is large enough. We also present a new fully sequential measurement strategy that addresses the challenge that nonconcavity it presents.

The Knowledge-Gradient Algorithm for Sequencing Experiments in Drug Discovery

We present a new technique for adaptively choosing the sequence of molecular compounds to test in drug discovery. Beginning with a base compound, we consider the problem of searching for a chemical derivative of the molecule that best treats a given disease. The problem of choosing molecules to test to maximize the expected quality of the best compound discovered may be formulated mathematically as a ranking-and-selection problem in which each molecule is an alternative. We apply a recently developed algorithm, known as the knowledge-gradient algorithm, that uses correlations in our Bayesian prior distribution between the performance of different alternatives (molecules) to dramatically reduce the number of molecular tests required, but it has heavy computational requirements that limit the number of possible alternatives to a few thousand. We develop computational improvements that allow the knowledge-gradient method to consider much larger sets of alternatives, and we demonstrate the method on a problem with 87,120 alternatives.

Sequential Sampling with Economics of Selection Procedures

Sequential sampling problems arise in stochastic simulation and many other applications. Sampling is used to infer the unknown performance of several alternatives before one alternative is selected as best. This paper presents new economically motivated fully sequential sampling procedures to solve such problems, called economics of selection procedures. The optimal procedure is derived for comparing a known standard with one alternative whose unknown reward is inferred with sampling. That result motivates heuristics when multiple alternatives have unknown rewards. The resulting procedures are more effective in numerical experiments than any previously proposed procedure of which we are aware and are easily implemented. The key driver of the improvement is the use of dynamic programming to model sequential sampling as an option to learn before selecting an alternative. It accounts for the expected benefit of adaptive stopping policies for sampling, rather than of one-stage policies, as is common in the literature. This paper was accepted by Assaf Zeevi, stochastic models and simulation.

Bisection Search with Noisy Responses

Bisection search is the most efficient algorithm for locating a unique point X ∗ ∈ (0, 1) when we are able to query an oracle only about whether X ∗ lies to the left or right of a point x of our choosing. We study a noisy version of this classic problem, where the oracles response is correct only with probability p. The probabilistic bisection algorithm (PBA) introduced by Horstein (IEEE Trans. Inform. Theory, 9 (1963), pp. 136-143) can be used to locate X ∗ in this setting. While the method works extremely well in practice, very little is known about its theoretical properties. In this paper, we provide several key findings about the PBA, which lead to the main conclusion that the expected absolute residuals of successive search results, i.e., E(|X ∗ − Xn|), converge to 0 at a geometric rate.

Sequential Bayes-Optimal Policies for Multiple Comparisons with a Known Standard

We consider the problem of efficiently allocating simulation effort to determine which of several simulated systems have mean performance exceeding a threshold of known value. Within a Bayesian formulation of this problem, the optimal fully sequential policy for allocating simulation effort is the solution to a dynamic program. When sampling is limited by probabilistic termination or sampling costs, we show that this dynamic program can be solved efficiently, providing a tractable way to compute the Bayes-optimal policy. The solution uses techniques from optimal stopping and multiarmed bandits. We then present further theoretical results characterizing this Bayes-optimal policy, compare it numerically to several approximate policies, and apply it to applications in emergency services and manufacturing.

The knowledge-gradient stopping rule for ranking and selection

We consider the ranking and selection of normal means in a fully sequential Bayesian context. By considering the sampling and stopping problems jointly rather than separately, we derive a new composite stopping/sampling rule. The sampling component of the derived composite rule is the same as the previously introduced LL1 sampling rule, but the stopping rule is new. This new stopping rule significantly improves the performance of LL1 as compared to its performance under the best other generally known adaptive stopping rule, EOC Bonf, outperforming it in every case tested.

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.

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. 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 real-world non-exchangeable data.