Peter I. Frazier

ICML Conference 2025 Conference Paper

Multi-Armed Bandits with Interference: Bridging Causal Inference and Adversarial Bandits

• Su Jia
• Peter I. Frazier
• Nathan Kallus

Experimentation with interference poses a significant challenge in contemporary online platforms. Prior research on experimentation with interference has concentrated on the final output of a policy. Cumulative performance, while equally important, is less well understood. To address this gap, we introduce the problem of Multi-armed Bandits with Interference (MABI), where the learner assigns an arm to each of $N$ experimental units over $T$ rounds. The reward of each unit in each round depends on the treatments of all units, where the interference between two units decays in their distance. The reward functions are chosen by an adversary and may vary arbitrarily over time and across different units. We first show that the optimal expected regret (against the best fixed-arm policy) is $\tilde O(\sqrt T)$, and can be achieved by a switchback policy. However, the regret (as a random variable) for any switchback policy suffers a high variance, since it does not account for $N$. We propose a policy based on a novel clustered randomization scheme, whose regret (i) is optimal in expectation and (ii) admits a high probability bound that vanishes in $N$.

ICML Conference 2024 Conference Paper

Bayesian Optimization of Function Networks with Partial Evaluations

• Poompol Buathong
• Jiayue Wan
• Raul Astudillo
• Samuel Daulton
• Maximilian Balandat
• Peter I. Frazier

Bayesian optimization is a powerful framework for optimizing functions that are expensive or time-consuming to evaluate. Recent work has considered Bayesian optimization of function networks (BOFN), where the objective function is given by a network of functions, each taking as input the output of previous nodes in the network as well as additional parameters. Leveraging this network structure has been shown to yield significant performance improvements. Existing BOFN algorithms for general-purpose networks evaluate the full network at each iteration. However, many real-world applications allow for evaluating nodes individually. To exploit this, we propose a novel knowledge gradient acquisition function that chooses which node and corresponding inputs to evaluate in a cost-aware manner, thereby reducing query costs by evaluating only on a part of the network at each step. We provide an efficient approach to optimizing our acquisition function and show that it outperforms existing BOFN methods and other benchmarks across several synthetic and real-world problems. Our acquisition function is the first to enable cost-aware optimization of a broad class of function networks.

NeurIPS Conference 2024 Conference Paper

Cost-aware Bayesian Optimization via the Pandora's Box Gittins Index

• Qian Xie
• Raul Astudillo
• Peter I. Frazier
• Ziv Scully
• Alexander Terenin

Bayesian optimization is a technique for efficiently optimizing unknown functions in a black-box manner. To handle practical settings where gathering data requires use of finite resources, it is desirable to explicitly incorporate function evaluation costs into Bayesian optimization policies. To understand how to do so, we develop a previously-unexplored connection between cost-aware Bayesian optimization and the Pandora's Box problem, a decision problem from economics. The Pandora's Box problem admits a Bayesian-optimal solution based on an expression called the Gittins index, which can be reinterpreted as an acquisition function. We study the use of this acquisition function for cost-aware Bayesian optimization, and demonstrate empirically that it performs well, particularly in medium-high dimensions. We further show that this performance carries over to classical Bayesian optimization without explicit evaluation costs. Our work constitutes a first step towards integrating techniques from Gittins index theory into Bayesian optimization.

ICML Conference 2023 Conference Paper

Smooth Non-stationary Bandits

• Su Jia
• Qian Xie 0005
• Nathan Kallus
• Peter I. Frazier

In many applications of online decision making, the environment is non-stationary and it is therefore crucial to use bandit algorithms that handle changes. Most existing approaches are designed to protect against non-smooth changes, constrained only by total variation or Lipschitzness over time, where they guarantee $T^{2/3}$ regret. However, in practice environments are often changing smoothly, so such algorithms may incur higher-than-necessary regret in these settings and do not leverage information on the rate of change. In this paper, we study a non-stationary two-arm bandit problem where we assume an arm’s mean reward is a $\beta$-Hölder function over (normalized) time, meaning it is $(\beta-1)$-times Lipschitz-continuously differentiable. We show the first separation between the smooth and non-smooth regimes by presenting a policy with $T^{3/5}$ regret for $\beta=2$. We complement this result by a $T^{\frac{\beta+1}{2\beta+1}}$ lower bound for any integer $\beta\ge 1$, which matches our upper bound for $\beta=2$.

ICML Conference 2019 Conference Paper

Bayesian Optimization of Composite Functions

• Raul Astudillo
• Peter I. Frazier

We consider optimization of composite objective functions, i. e. , of the form $f(x)=g(h(x))$, where $h$ is a black-box derivative-free expensive-to-evaluate function with vector-valued outputs, and $g$ is a cheap-to-evaluate real-valued function. While these problems can be solved with standard Bayesian optimization, we propose a novel approach that exploits the composite structure of the objective function to substantially improve sampling efficiency. Our approach models $h$ using a multi-output Gaussian process and chooses where to sample using the expected improvement evaluated on the implied non-Gaussian posterior on $f$, which we call expected improvement for composite functions (EI-CF). Although EI-CF cannot be computed in closed form, we provide a novel stochastic gradient estimator that allows its efficient maximization. We also show that our approach is asymptotically consistent, i. e. , that it recovers a globally optimal solution as sampling effort grows to infinity, generalizing previous convergence results for classical expected improvement. Numerical experiments show that our approach dramatically outperforms standard Bayesian optimization benchmarks, reducing simple regret by several orders of magnitude.

UAI Conference 2019 Conference Paper

Practical Multi-fidelity Bayesian Optimization for Hyperparameter Tuning

• Jian Wu
• Saul Toscano-Palmerin
• Peter I. Frazier
• Andrew Gordon Wilson

Bayesian optimization is popular for optimizing time-consuming black-box objectives. Nonetheless, for hyperparameter tuning in deep neural networks, the time required to evaluate the validation error for even a few hyperparameter settings remains a bottleneck. Multi-fidelity optimization promises relief using cheaper proxies to such objectives — for example, validation error for a network trained using a subset of the training points or fewer iterations than required for convergence. We propose a highly flexible and practical approach to multi-fidelity Bayesian optimization, focused on efficiently optimizing hyperparameters for iteratively trained supervised learning models. We introduce a new acquisition function, the trace-aware knowledge-gradient, which efficiently leverages both multiple continuous fidelity controls and trace observations — values of the objective at a sequence of fidelities, available when varying fidelity using training iterations. We provide a provably convergent method for optimizing our acquisition function and show it outperforms state-of-the-art alternatives for hyperparameter tuning of deep neural networks and large-scale kernel learning.

ICML Conference 2017 Conference Paper

Dueling Bandits with Weak Regret

• Bangrui Chen
• Peter I. Frazier

We consider online content recommendation with implicit feedback through pairwise comparisons, formalized as the so-called dueling bandit problem. We study the dueling bandit problem in the Condorcet winner setting, and consider two notions of regret: the more well-studied strong regret, which is 0 only when both arms pulled are the Condorcet winner; and the less well-studied weak regret, which is 0 if either arm pulled is the Condorcet winner. We propose a new algorithm for this problem, Winner Stays (WS), with variations for each kind of regret: WS for weak regret (WS-W) has expected cumulative weak regret that is $O(N^2)$, and $O(N\log(N))$ if arms have a total order; WS for strong regret (WS-S) has expected cumulative strong regret of $O(N^2 + N \log(T))$, and $O(N\log(N)+N\log(T))$ if arms have a total order. WS-W is the first dueling bandit algorithm with weak regret that is constant in time. WS is simple to compute, even for problems with many arms, and we demonstrate through numerical experiments on simulated and real data that WS has significantly smaller regret than existing algorithms in both the weak- and strong-regret settings.

ICML Conference 2015 Conference Paper

Bayesian Multiple Target Localization

• Purnima Rajan
• Weidong Han 0004
• Raphael Sznitman
• Peter I. Frazier
• Bruno Jedynak

We consider the problem of quickly localizing multiple targets by asking questions of the form “How many targets are within this set" while obtaining noisy answers. This setting is a generalization to multiple targets of the game of 20 questions in which only a single target is queried. We assume that the targets are points on the real line, or in a two dimensional plane for the experiments, drawn independently from a known distribution. We evaluate the performance of a policy using the expected entropy of the posterior distribution after a fixed number of questions with noisy answers. We derive a lower bound for the value of this problem and study a specific policy, named the dyadic policy. We show that this policy achieves a value which is no more than twice this lower bound when answers are noise-free, and show a more general constant factor approximation guarantee for the noisy setting. We present an empirical evaluation of this policy on simulated data for the problem of detecting multiple instances of the same object in an image. Finally, we present experiments on localizing multiple faces simultaneously on real images.

ICML Conference 2013 Conference Paper

An Optimal Policy for Target Localization with Application to Electron Microscopy

• Raphael Sznitman
• Aurélien Lucchi
• Peter I. Frazier
• Bruno Jedynak
• Pascal Fua

This paper considers the task of finding a target location by making a limited number of sequential observations. Each observation results from evaluating an imperfect classifier of a chosen cost and accuracy on an interval of chosen length and position. Within a Bayesian framework, we study the problem of minimizing an objective that combines the entropy of the posterior distribution with the cost of the questions asked. In this problem, we show that the one-step lookahead policy is Bayes-optimal for any arbitrary time horizon. Moreover, this one-step lookahead policy is easy to compute and implement. We then use this policy in the context of localizing mitochondria in electron microscope images, and experimentally show that significant speed ups in acquisition can be gained, while maintaining near equal image quality at target locations, when compared to current policies.

JMLR Journal 2011 Journal Article

Distance Dependent Chinese Restaurant Processes

• David M. Blei
• Peter I. Frazier

We develop the distance dependent Chinese restaurant process, a flexible class of distributions over partitions that allows for dependencies between the elements. This class can be used to model many kinds of dependencies between data in infinite clustering models, including dependencies arising from time, space, and network connectivity. We examine the properties of the distance dependent CRP, discuss its connections to Bayesian nonparametric mixture models, and derive a Gibbs sampler for both fully observed and latent mixture settings. We study its empirical 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 and network data. We also show that the distance dependent CRP representation of the traditional CRP mixture leads to a faster-mixing Gibbs sampling algorithm than the one based on the original formulation. [abs] [ pdf ][ bib ] &copy JMLR 2011. ( edit, beta )

JMLR Journal 2011 Journal Article

Hierarchical Knowledge Gradient for Sequential Sampling

• Martijn R.K. Mes
• Warren B. Powell
• Peter I. Frazier

We propose a sequential sampling policy for noisy discrete global optimization and ranking and selection, in which we aim to efficiently explore a finite set of alternatives before selecting an alternative as best when exploration stops. Each alternative may be characterized by a multi-dimensional vector of categorical and numerical attributes and has independent normal rewards. We use a Bayesian probability model for the unknown reward of each alternative and follow a fully sequential sampling policy called the knowledge-gradient policy. This policy myopically optimizes the expected increment in the value of sampling information in each time period. We propose a hierarchical aggregation technique that uses the common features shared by alternatives to learn about many alternatives from even a single measurement. This approach greatly reduces the measurement effort required, but it requires some prior knowledge on the smoothness of the function in the form of an aggregation function and computational issues limit the number of alternatives that can be easily considered to the thousands. We prove that our policy is consistent, finding a globally optimal alternative when given enough measurements, and show through simulations that it performs competitively with or significantly better than other policies. [abs] [ pdf ][ bib ] &copy JMLR 2011. ( edit, beta )

ICML Conference 2010 Conference Paper

Distance dependent Chinese restaurant processes

• David M. Blei
• Peter I. Frazier