Residual Bootstrap Exploration for Bandit Algorithms
Department of Statistics, Purdue University（普渡大学统计系）；Department of Electrical Engineering, Princeton University（普林斯顿大学电气工程系）；Purdue University（普渡大学）；Princeton University（普林斯顿大学）
In this paper, we propose a novel perturbation-based exploration method in bandit algorithms with bounded or unbounded rewards, called residual bootstrap exploration (\texttt{ReBoot}). The \texttt{ReBoot} enforces exploration by injecting data-driven randomness through a residual-based perturbation mechanism. This novel mechanism captures the underlying distributional properties of fitting errors, and more importantly boosts exploration to escape from suboptimal solutions (for small sample sizes) by inflating variance level in an \textit{unconventional} way. In theory, with appropriate variance inflation level, \texttt{ReBoot} provably secures instance-dependent logarithmic regret in Gaussian multi-armed bandits. We evaluate the \texttt{ReBoot} in different synthetic multi-armed bandits problems and observe that the \texttt{ReBoot} performs better for unbounded rewards and more robustly than \texttt{Giro} \cite{kveton2018garbage} and \texttt{PHE} \cite{kveton2019perturbed}, with comparable computational efficiency to the Thompson sampling method.
University of California, Los Angeles（加州大学洛杉矶分校）；Purdue University（普渡大学）
We propose a new bootstrap-based online algorithm for stochastic linear bandit problems. The key idea is to adopt residual bootstrap exploration, in which the agent estimates the next step reward by re-sampling the residuals of mean reward estimate. Our algorithm, residual bootstrap exploration for stochastic linear bandit (\texttt{LinReBoot}), estimates the linear reward from its re-sampling distribution and pulls the arm with the highest reward estimate. In particular, we contribute a theoretical framework to demystify residual bootstrap-based exploration mechanisms in stochastic linear band…
University of British Columbia（英属哥伦比亚大学）；Adobe Research（Adobe研究）
We investigate the use of bootstrapping in the bandit setting. We first show that the commonly used non-parametric bootstrapping (NPB) procedure can be provably inefficient and establish a near-linear lower bound on the regret incurred by it under the bandit model with Bernoulli rewards. We show that NPB with an appropriate amount of forced exploration can result in sub-linear albeit sub-optimal regret. As an alternative to NPB, we propose a weighted bootstrapping (WB) procedure. For Bernoulli rewards, WB with multiplicative exponential weights is mathematically equivalent to Thompson sampling…
Indian Institute of Technology, Madras（印度理工学院马德拉斯分校）
This study presents two new algorithms for solving linear stochastic bandit problems. The proposed methods use an approach from non-parametric statistics called bootstrapping to create confidence bounds. This is achieved without making any assumptions about the distribution of noise in the underlying system. We present the X-Random and X-Fixed bootstrap bandits which correspond to the two well-known approaches for conducting bootstraps on models, in the literature. The proposed methods are compared to other popular solutions for linear stochastic bandit problems, namely, OFUL, LinUCB and Thomp…
Google Research（谷歌研究院）；DeepMind（深度思维）；University of Alberta（阿尔伯塔大学）；Mila, University of Montreal（蒙特利尔大学米拉研究所）；Adobe Research（奥多比研究院）；Facebook AI Research（脸书人工智能研究院）
We propose a bandit algorithm that explores by randomizing its history of rewards. Specifically, it pulls the arm with the highest mean reward in a non-parametric bootstrap sample of its history with pseudo rewards. We design the pseudo rewards such that the bootstrap mean is optimistic with a sufficiently high probability. We call our algorithm Giro, which stands for garbage in, reward out. We analyze Giro in a Bernoulli bandit and derive a $O(K Δ^{-1} \log n)$ bound on its $n$-round regret, where $Δ$ is the difference in the expected rewards of the optimal and the best suboptimal arms, and $…
This work explores adaptations of successful multi-armed bandits policies to the online contextual bandits scenario with binary rewards using binary classification algorithms such as logistic regression as black-box oracles. Some of these adaptations are achieved through bootstrapping or approximate bootstrapping, while others rely on other forms of randomness, resulting in more scalable approaches than previous works, and the ability to work with any type of classification algorithm. In particular, the Adaptive-Greedy algorithm shows a lot of promise, in many cases achieving better performanc…
Thompson sampling provides a solution to bandit problems in which new observations are allocated to arms with the posterior probability that an arm is optimal. While sometimes easy to implement and asymptotically optimal, Thompson sampling can be computationally demanding in large scale bandit problems, and its performance is dependent on the model fit to the observed data. We introduce bootstrap Thompson sampling (BTS), a heuristic method for solving bandit problems which modifies Thompson sampling by replacing the posterior distribution used in Thompson sampling by a bootstrap distribution. …
University of Lille（里尔大学）；CNRS（法国国家科学研究中心）；INRIA Lille Nord Europe（法国国家信息与自动化研究所里尔北欧洲分部）
In many recommendation applications such as news recommendation, the items that can be rec- ommended come and go at a very fast pace. This is a challenge for recommender systems (RS) to face this setting. Online learning algorithms seem to be the most straight forward solution. The contextual bandit framework was introduced for that very purpose. In general the evaluation of a RS is a critical issue. Live evaluation is of- ten avoided due to the potential loss of revenue, hence the need for offline evaluation methods. Two options are available. Model based meth- ods are biased by nature and ar…
Amazon（亚马逊）；North Carolina State University（北卡罗来纳州立大学）
Despite the great interest in the bandit problem, designing efficient algorithms for complex models remains challenging, as there is typically no analytical way to quantify uncertainty. In this paper, we propose Multiplier Bootstrap-based Exploration (MBE), a novel exploration strategy that is applicable to any reward model amenable to weighted loss minimization. We prove both instance-dependent and instance-independent rate-optimal regret bounds for MBE in sub-Gaussian multi-armed bandits. With extensive simulation and real data experiments, we show the generality and adaptivity of MBE.
Institute of Computing Technology, Chinese Academy of Sciences（中国科学院计算技术研究所）；University of Chinese Academy of Sciences（中国科学院大学）；Tencent（腾讯）
The exploration/exploitation (E&E) dilemma lies at the core of interactive systems such as online advertising, for which contextual bandit algorithms have been proposed. Bayesian approaches provide guided exploration with principled uncertainty estimation, but the applicability is often limited due to over-simplified assumptions. Non-Bayesian bootstrap methods, on the other hand, can apply to complex problems by using deep reward models, but lacks clear guidance to the exploration behavior. It still remains largely unsolved to develop a practical method for complex deep contextual bandits. In …
This technical note presents a new approach to carrying out the kind of exploration achieved by Thompson sampling, but without explicitly maintaining or sampling from posterior distributions. The approach is based on a bootstrap technique that uses a combination of observed and artificially generated data. The latter serves to induce a prior distribution which, as we will demonstrate, is critical to effective exploration. We explain how the approach can be applied to multi-armed bandit and reinforcement learning problems and how it relates to Thompson sampling. The approach is particularly wel…
Columbia University（哥伦比亚大学）；Moloco（Moloco公司）；University of New Hampshire（新罕布什尔大学）
Many efficient algorithms with strong theoretical guarantees have been proposed for the contextual multi-armed bandit problem. However, applying these algorithms in practice can be difficult because they require domain expertise to build appropriate features and to tune their parameters. We propose a new method for the contextual bandit problem that is simple, practical, and can be applied with little or no domain expertise. Our algorithm relies on decision trees to model the context-reward relationship. Decision trees are non-parametric, interpretable, and work well without hand-crafted featu…
Netflix；Cornell University（康奈尔大学）；Cornell University & Netflix（康奈尔大学 & Netflix）
Exploration is a crucial aspect of bandit and reinforcement learning algorithms. The uncertainty quantification necessary for exploration often comes from either closed-form expressions based on simple models or resampling and posterior approximations that are computationally intensive. We propose instead an approximate exploration methodology based on fitting only two point estimates, one tuned and one overfit. The approach, which we term the residual overfit method of exploration (ROME), drives exploration towards actions where the overfit model exhibits the most overfitting compared to the …
Amazon Customer Service provides real-time support for millions of customer contacts every year. While bot-resolver helps automate some traffic, we still see high demand for human agents, also called subject matter experts (SMEs). Customers outreach with questions in different domains (return policy, device troubleshooting, etc.). Depending on their training, not all SMEs are eligible to handle all contacts. Routing contacts to eligible SMEs turns out to be a non-trivial problem because SMEs' domain eligibility is subject to training quality and can change over time. To optimally recommend SME…
University of California, San Diego（加利福尼亚大学圣迭戈分校）；Twitter（推特公司）
Recommender systems trained in a continuous learning fashion are plagued by the feedback loop problem, also known as algorithmic bias. This causes a newly trained model to act greedily and favor items that have already been engaged by users. This behavior is particularly harmful in personalised ads recommendations, as it can also cause new campaigns to remain unexplored. Exploration aims to address this limitation by providing new information about the environment, which encompasses user preference, and can lead to higher long-term reward. In this work, we formulate a display advertising recom…
University of Toronto（多伦多大学）；The Chinese University of Hong Kong（香港中文大学）
It has been recently shown in the literature that the sample averages from online learning experiments are biased when used to estimate the mean reward. To correct the bias, off-policy evaluation methods, including importance sampling and doubly robust estimators, typically calculate the conditional propensity score, which is ill-defined for non-randomized policies such as UCB. This paper provides a procedure to debias the samples using bootstrap, which doesn't require the knowledge of the reward distribution and can be applied to any adaptive policies. Numerical experiments demonstrate the ef…
This paper examines the use of a residual bootstrap for bias correction in machine learning regression methods. Accounting for bias is an important obstacle in recent efforts to develop statistical inference for machine learning methods. We demonstrate empirically that the proposed bootstrap bias correction can lead to substantial improvements in both bias and predictive accuracy. In the context of ensembles of trees, we show that this correction can be approximated at only double the cost of training the original ensemble without introducing additional variance. Our method is shown to improve…
Columbia University（哥伦比亚大学）；University of Pennsylvania（宾夕法尼亚大学）
The stochastic $K$-armed bandit problem has been studied extensively due to its applications in various domains ranging from online advertising to clinical trials. In practice however, the number of arms can be very large resulting in large memory requirements for simultaneously processing them. In this paper we consider a streaming setting where the arms are presented in a stream and the algorithm uses limited memory to process these arms. Here, the goal is not only to minimize regret, but also to do so in minimal memory. Previous algorithms for this problem operate in one of the two settings…
The Institute for Computational and Mathematical Engineering（计算与数学工程研究所）；Stanford University（斯坦福大学）；Meta Platforms, Inc.（Meta平台公司）；Ads Online Experimentation（广告在线实验部）；Central Applied Science（应用科学中央研究所）
Tsinghua University（清华大学）；Stanford University（斯坦福大学）；University of Washington（华盛顿大学）
This paper presents a new model-free algorithm for episodic finite-horizon Markov Decision Processes (MDP), Adaptive Multi-step Bootstrap (AMB), which enjoys a stronger gap-dependent regret bound. The first innovation is to estimate the optimal $Q$-function by combining an optimistic bootstrap with an adaptive multi-step Monte Carlo rollout. The second innovation is to select the action with the largest confidence interval length among admissible actions that are not dominated by any other actions. We show when each state has a unique optimal action, AMB achieves a gap-dependent regret bound t…
We propose Streaming Bandits, a Restless Multi Armed Bandit (RMAB) framework in which heterogeneous arms may arrive and leave the system after staying on for a finite lifetime. Streaming Bandits naturally capture the health intervention planning problem, where health workers must manage the health outcomes of a patient cohort while new patients join and existing patients leave the cohort each day. Our contributions are as follows: (1) We derive conditions under which our problem satisfies indexability, a precondition that guarantees the existence and asymptotic optimality of the Whittle Index …
Motivated by the application of real-time pricing in e-commerce platforms, we consider the problem of revenue-maximization in a setting where the seller can leverage contextual information describing the customer's history and the product's type to predict her valuation of the product. However, her true valuation is unobservable to the seller, only binary outcome in the form of success-failure of a transaction is observed. Unlike in usual contextual bandit settings, the optimal price/arm given a covariate in our setting is sensitive to the detailed characteristics of the residual uncertainty d…
Diffusion Approximations for Thompson Sampling in the Small Gap Regime
Kellogg School of Management, Northwestern University（西北大学凯洛格管理学院）；Department of Management Science and Engineering, Stanford University（斯坦福大学管理科学与工程系）
Distribution forecast can quantify forecast uncertainty and provide various forecast scenarios with their corresponding estimated probabilities. Accurate distribution forecast is crucial for planning - for example when making production capacity or inventory allocation decisions. We propose a practical and robust distribution forecast framework that relies on backtest-based bootstrap and adaptive residual selection. The proposed approach is robust to the choice of the underlying forecasting model, accounts for uncertainty around the input covariates, and relaxes the independence between residu…
DOLCE: Decomposing Off-Policy Evaluation/Learning into Lagged and Current Effects
Department of Multidisciplinary Sciences, Graduate School of Arts and Sciences, The University of Tokyo（东京大学艺术及科学研究生院多学科科学系）；Department of Epidemiology, National Institute of Infectious Diseases, Japan Institute for Health Security（日本国立感染症研究所健康安全研究所流行病学系）
Tencent AI Lab（腾讯人工智能实验室）；The Pennsylvania State University（宾夕法尼亚州立大学）
Offline reinforcement learning (RL) tries to learn the near-optimal policy with recorded offline experience without online exploration. Current offline RL research includes: 1) generative modeling, i.e., approximating a policy using fixed data; and 2) learning the state-action value function. While most research focuses on the state-action function part through reducing the bootstrapping error in value function approximation induced by the distribution shift of training data, the effects of error propagation in generative modeling have been neglected. In this paper, we analyze the error in gen…
While in general trading off exploration and exploitation in reinforcement learning is hard, under some formulations relatively simple solutions exist. In this paper, we first derive upper bounds for the utility of selecting different actions in the multi-armed bandit setting. Unlike the common statistical upper confidence bounds, these explicitly link the planning horizon, uncertainty and the need for exploration explicit. The resulting algorithm can be seen as a generalisation of the classical Thompson sampling algorithm. We experimentally test these algorithms, as well as $ε$-greedy and the…
Laboratory for Information and Decision Systems（信息与决策系统实验室）；Statistics and Data Science Center（统计与数据科学中心）；EECS and Mathematics（电子工程与科学系及数学）；Massachusetts Institute of Technology（麻省理工学院）
We study the residual bootstrap (RB) method in the context of high-dimensional linear regression. Specifically, we analyze the distributional approximation of linear contrasts $c^{\top} (\hatβ_ρ-β)$, where $\hatβ_ρ$ is a ridge-regression estimator. When regression coefficients are estimated via least squares, classical results show that RB consistently approximates the laws of contrasts, provided that $p\ll n$, where the design matrix is of size $n\times p$. Up to now, relatively little work has considered how additional structure in the linear model may extend the validity of RB to the settin…
