We introduce the bilinear bandit problem with low-rank structure in which an action takes the form of a pair of arms from two different entity types, and the reward is a bilinear function of the known feature vectors of the arms. The unknown in the problem is a $d_1$ by $d_2$ matrix $\mathbfΘ^*$ that defines the reward, and has low rank $r \ll \min\{d_1,d_2\}$. Determination of $\mathbfΘ^*$ with this low-rank structure poses a significant challenge in finding the right exploration-exploitation tradeoff. In this work, we propose a new two-stage algorithm called "Explore-Subspace-Then-Refine" (ESTR). The first stage is an explicit subspace exploration, while the second stage is a linear bandit algorithm called "almost-low-dimensional OFUL" (LowOFUL) that exploits and further refines the estimated subspace via a regularization technique. We show that the regret of ESTR is $\widetilde{\mathcal{O}}((d_1+d_2)^{3/2} \sqrt{r T})$ where $\widetilde{\mathcal{O}}$ hides logarithmic factors and $T$ is the time horizon, which improves upon the regret of $\widetilde{\mathcal{O}}(d_1d_2\sqrt{T})$ attained for a naïve linear bandit reduction. We conjecture that the regret bound of ESTR is unimprovable up to polylogarithmic factors, and our preliminary experiment shows that ESTR outperforms a naïve linear bandit reduction.
PSL - Université Paris Dauphine（巴黎多芬大学（PSL研究大学））；CNRS（法国国家科学研究中心）；Huawei Noah’s Ark Lab（华为诺亚方舟实验室）
We propose the first regret-based approach to the Graphical Bilinear Bandits problem, where $n$ agents in a graph play a stochastic bilinear bandit game with each of their neighbors. This setting reveals a combinatorial NP-hard problem that prevents the use of any existing regret-based algorithm in the (bi-)linear bandit literature. In this paper, we fill this gap and present the first regret-based algorithm for graphical bilinear bandits using the principle of optimism in the face of uncertainty. Theoretical analysis of this new method yields an upper bound of $\tilde{O}(\sqrt{T})$ on the $α$…
PSL - Université Paris Dauphine（巴黎文理研究大学-巴黎多芬大学）；CNRS（法国国家科学研究中心）；Huawei Noah’s Ark Lab（华为诺亚方舟实验室）；Liverpool University（利物浦大学）
We introduce a new graphical bilinear bandit problem where a learner (or a \emph{central entity}) allocates arms to the nodes of a graph and observes for each edge a noisy bilinear reward representing the interaction between the two end nodes. We study the best arm identification problem in which the learner wants to find the graph allocation maximizing the sum of the bilinear rewards. By efficiently exploiting the geometry of this bandit problem, we propose a \emph{decentralized} allocation strategy based on random sampling with theoretical guarantees. In particular, we characterize the influ…
Efficient Uncoupled Learning Dynamics with $\tilde{O}\!\left(T^{-1/4}\right)$ Last-Iterate Convergence in Bilinear Saddle-Point Problems over Convex Sets under Bandit Feedback
University of Washington（华盛顿大学）；University of California, Irvine（加州大学尔湾分校）
ENS Paris-Saclay（巴黎萨克雷高等师范学院）；CNRS（法国国家科学研究中心）；Université de Lille（里尔大学）；Inria（法国国家信息与自动化研究所）；DeepMind（深度思维）；CentraleSupélec（中央理工-高等电力学院）
Stochastic Rank-One Bandits (Katarya et al, (2017a,b)) are a simple framework for regret minimization problems over rank-one matrices of arms. The initially proposed algorithms are proved to have logarithmic regret, but do not match the existing lower bound for this problem. We close this gap by first proving that rank-one bandits are a particular instance of unimodal bandits, and then providing a new analysis of Unimodal Thompson Sampling (UTS), initially proposed by Paladino et al (2017). We prove an asymptotically optimal regret bound on the frequentist regret of UTS and we support our clai…
Personalization is a crucial aspect of many online experiences. In particular, content ranking is often a key component in delivering sophisticated personalization results. Commonly, supervised learning-to-rank methods are applied, which suffer from bias introduced during data collection by production systems in charge of producing the ranking. To compensate for this problem, we leverage contextual multi-armed bandits. We propose novel extensions of two well-known algorithms viz. LinUCB and Linear Thompson Sampling to the ranking use-case. To account for the biases in a production environment,…
MIT CSAIL（麻省理工学院计算机科学与人工智能实验室）；Microsoft Research（微软研究院）；Chicago Booth（芝加哥大学布斯商学院）
We consider dynamic pricing with many products under an evolving but low-dimensional demand model. Assuming the temporal variation in cross-elasticities exhibits low-rank structure based on fixed (latent) features of the products, we show that the revenue maximization problem reduces to an online bandit convex optimization with side information given by the observed demands. We design dynamic pricing algorithms whose revenue approaches that of the best fixed price vector in hindsight, at a rate that only depends on the intrinsic rank of the demand model and not the number of products. Our appr…
This paper studies the problem of adaptively sampling from K distributions (arms) in order to identify the largest gap between any two adjacent means. We call this the MaxGap-bandit problem. This problem arises naturally in approximate ranking, noisy sorting, outlier detection, and top-arm identification in bandits. The key novelty of the MaxGap-bandit problem is that it aims to adaptively determine the natural partitioning of the distributions into a subset with larger means and a subset with smaller means, where the split is determined by the largest gap rather than a pre-specified rank or t…
In a low-rank linear bandit problem, the reward of an action (represented by a matrix of size $d_1 \times d_2$) is the inner product between the action and an unknown low-rank matrix $Θ^*$. We propose an algorithm based on a novel combination of online-to-confidence-set conversion~\citep{abbasi2012online} and the exponentially weighted average forecaster constructed by a covering of low-rank matrices. In $T$ rounds, our algorithm achieves $\widetilde{O}((d_1+d_2)^{3/2}\sqrt{rT})$ regret that improves upon the standard linear bandit regret bound of $\widetilde{O}(d_1d_2\sqrt{T})$ when the rank …
Non-stationarity appears in many online applications such as web search and advertising. In this paper, we study the online learning to rank problem in a non-stationary environment where user preferences change abruptly at an unknown moment in time. We consider the problem of identifying the K most attractive items and propose cascading non-stationary bandits, an online learning variant of the cascading model, where a user browses a ranked list from top to bottom and clicks on the first attractive item. We propose two algorithms for solving this non-stationary problem: CascadeDUCB and CascadeS…
Stochastic Low-rank Tensor Bandits for Multi-dimensional Online Decision Making
Amazon（亚马逊）；Deepmind（深度思维）；Miami Herbert Business School, University of Miami（迈阿密大学迈阿密赫伯特商学院）；Krannert School of Management, Purdue University（普渡大学克兰纳特管理学院）
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；Adobe Research（Adobe研究院）；University of Alberta（阿尔伯塔大学）；Telecom ParisTech（巴黎高等电信学院）
The probability that a user will click a search result depends both on its relevance and its position on the results page. The position based model explains this behavior by ascribing to every item an attraction probability, and to every position an examination probability. To be clicked, a result must be both attractive and examined. The probabilities of an item-position pair being clicked thus form the entries of a rank-$1$ matrix. We propose the learning problem of a Bernoulli rank-$1$ bandit where at each step, the learning agent chooses a pair of row and column arms, and receives the prod…
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；Adobe Research（Adobe研究院）；University of Alberta（阿尔伯塔大学）；Telecom ParisTech（巴黎高等电信学院）
We propose stochastic rank-$1$ bandits, a class of online learning problems where at each step a learning agent chooses a pair of row and column arms, and receives the product of their values as a reward. The main challenge of the problem is that the individual values of the row and column are unobserved. We assume that these values are stochastic and drawn independently. We propose a computationally-efficient algorithm for solving our problem, which we call Rank1Elim. We derive a $O((K + L) (1 / Δ) \log n)$ upper bound on its $n$-step regret, where $K$ is the number of rows, $L$ is the number…
California Institute of Technology（加州理工学院）；University of California, Irvine（加州大学欧文分校）
High-dimensional representations often have a lower dimensional underlying structure. This is particularly the case in many decision making settings. For example, when the representation of actions is generated from a deep neural network, it is reasonable to expect a low-rank structure whereas conventional structures like sparsity are not valid anymore. Subspace recovery methods, such as Principle Component Analysis (PCA) can find the underlying low-rank structures in the feature space and reduce the complexity of the learning tasks. In this work, we propose Projected Stochastic Linear Bandit …
University of Copenhagen（哥本哈根大学）；University College London（伦敦大学学院）
New ranking algorithms are continually being developed and refined, necessitating the development of efficient methods for evaluating these rankers. Online ranker evaluation focuses on the challenge of efficiently determining, from implicit user feedback, which ranker out of a finite set of rankers is the best. Online ranker evaluation can be modeled by dueling ban- dits, a mathematical model for online learning under limited feedback from pairwise comparisons. Comparisons of pairs of rankers is performed by interleaving their result sets and examining which documents users click on. The dueli…
Adobe Research（Adobe研究院）；University of Alberta（阿尔伯塔大学）；Yahoo Labs（雅虎实验室）；Technicolor Research（特艺集团研究院）
A search engine usually outputs a list of $K$ web pages. The user examines this list, from the first web page to the last, and chooses the first attractive page. This model of user behavior is known as the cascade model. In this paper, we propose cascading bandits, a learning variant of the cascade model where the objective is to identify $K$ most attractive items. We formulate our problem as a stochastic combinatorial partial monitoring problem. We propose two algorithms for solving it, CascadeUCB1 and CascadeKL-UCB. We also prove gap-dependent upper bounds on the regret of these algorithms a…
Peking University（北京大学）；University of California, Berkeley（加州大学伯克利分校）；Facebook AI Research（脸书人工智能研究院）；University of Washington（华盛顿大学）
In this paper, we present a reduction-based framework for conservative bandits and RL, in which our core technique is to calculate the necessary and sufficient budget obtained from running the baseline policy. For lower bounds, we improve the existing lower bound for conservative multi-armed bandits and obtain new lower bounds for conservative linear bandits, tabular RL and low-rank MDP, through a black-box reduction that turns a certain lower bound in the nonconservative setting into a new lower bound in the conservative setting. For upper bounds, in multi-armed bandits, linear bandits and ta…
Adobe Research（Adobe 研究院）；University of Alberta（阿尔伯塔大学）；Rutgers University（罗格斯大学）
Many problems in computer vision and recommender systems involve low-rank matrices. In this work, we study the problem of finding the maximum entry of a stochastic low-rank matrix from sequential observations. At each step, a learning agent chooses pairs of row and column arms, and receives the noisy product of their latent values as a reward. The main challenge is that the latent values are unobserved. We identify a class of non-negative matrices whose maximum entry can be found statistically efficiently and propose an algorithm for finding them, which we call LowRankElim. We derive a $\Decla…
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；Adobe Research（奥多比研究院）；University of Alberta（阿尔伯塔大学）
A search engine recommends to the user a list of web pages. The user examines this list, from the first page to the last, and clicks on all attractive pages until the user is satisfied. This behavior of the user can be described by the dependent click model (DCM). We propose DCM bandits, an online learning variant of the DCM where the goal is to maximize the probability of recommending satisfactory items, such as web pages. The main challenge of our learning problem is that we do not observe which attractive item is satisfactory. We propose a computationally-efficient learning algorithm for so…
Meta AI；CREST, ENSAE（法国国立统计与经济管理学院经济与统计研究中心）；Core Data Science, Meta（Meta核心数据科学部）
We consider a multi-armed bandit setting where, at the beginning of each round, the learner receives noisy independent, and possibly biased, \emph{evaluations} of the true reward of each arm and it selects $K$ arms with the objective of accumulating as much reward as possible over $T$ rounds. Under the assumption that at each round the true reward of each arm is drawn from a fixed distribution, we derive different algorithmic approaches and theoretical guarantees depending on how the evaluations are generated. First, we show a $\widetilde{O}(T^{2/3})$ regret in the general case when the observ…
University of Amsterdam（阿姆斯特丹大学）；Bloomberg（彭博）；Ahold Delhaize（阿霍德·德尔海兹）
Relevance ranking and result diversification are two core areas in modern recommender systems. Relevance ranking aims at building a ranked list sorted in decreasing order of item relevance, while result diversification focuses on generating a ranked list of items that covers a broad range of topics. In this paper, we study an online learning setting that aims to recommend a ranked list with $K$ items that maximizes the ranking utility, i.e., a list whose items are relevant and whose topics are diverse. We formulate it as the cascade hybrid bandits (CHB) problem. CHB assumes the cascading user …
Indian Institute of Technology (BHU) Varanasi（印度理工学院（巴纳拉斯印度教大学）瓦拉纳西分校）；Yahoo Japan Corporation（雅虎日本公司）
We propose an extensible deep learning method that uses reinforcement learning to train neural networks for offline ranking in information retrieval (IR). We call our method BanditRank as it treats ranking as a contextual bandit problem. In the domain of learning to rank for IR, current deep learning models are trained on objective functions different from the measures they are evaluated on. Since most evaluation measures are discrete quantities, they cannot be leveraged by directly using gradient descent algorithms without an approximation. BanditRank bridges this gap by directly optimizing a…
We consider combinatorial online learning with subset choices when only relative feedback information from subsets is available, instead of bandit or semi-bandit feedback which is absolute. Specifically, we study two regret minimisation problems over subsets of a finite ground set $[n]$, with subset-wise relative preference information feedback according to the Multinomial logit choice model. In the first setting, the learner can play subsets of size bounded by a maximum size and receives top-$m$ rank-ordered feedback, while in the second setting the learner can play subsets of a fixed size $k…
Microsoft Research Silicon Valley（微软研究院硅谷分院）；Microsoft Research Cambridge（微软研究院剑桥分院）
Most learning to rank research has assumed that the utility of different documents is independent, which results in learned ranking functions that return redundant results. The few approaches that avoid this have rather unsatisfyingly lacked theoretical foundations, or do not scale. We present a learning-to-rank formulation that optimizes the fraction of satisfied users, with several scalable algorithms that explicitly takes document similarity and ranking context into account. Our formulation is a non-trivial common generalization of two multi-armed bandit models from the literature: "ranked …
University of Washington（华盛顿大学）；Tsinghua University（清华大学）；Peking University（北京大学）；Princeton University（普林斯顿大学）；Microsoft Research（微软研究院）
Bandit problems with linear or concave reward have been extensively studied, but relatively few works have studied bandits with non-concave reward. This work considers a large family of bandit problems where the unknown underlying reward function is non-concave, including the low-rank generalized linear bandit problems and two-layer neural network with polynomial activation bandit problem. For the low-rank generalized linear bandit problem, we provide a minimax-optimal algorithm in the dimension, refuting both conjectures in [LMT21, JWWN19]. Our algorithms are based on a unified zeroth-order o…
We introduce the factored bandits model, which is a framework for learning with limited (bandit) feedback, where actions can be decomposed into a Cartesian product of atomic actions. Factored bandits incorporate rank-1 bandits as a special case, but significantly relax the assumptions on the form of the reward function. We provide an anytime algorithm for stochastic factored bandits and up to constants matching upper and lower regret bounds for the problem. Furthermore, we show that with a slight modification the proposed algorithm can be applied to utility based dueling bandits. We obtain an …
Stochastic high dimensional bandit problems with low dimensional structures are useful in different applications such as online advertising and drug discovery. In this work, we propose a simple unified algorithm for such problems and present a general analysis framework for the regret upper bound of our algorithm. We show that under some mild unified assumptions, our algorithm can be applied to different high dimensional bandit problems. Our framework utilizes the low dimensional structure to guide the parameter estimation in the problem, therefore our algorithm achieves the comparable regret …
Carnegie Mellon University（卡内基梅隆大学）；New York University（纽约大学）
The Thresholding Bandit Problem (TBP) aims to find the set of arms with mean rewards greater than a given threshold. We consider a new setting of TBP, where in addition to pulling arms, one can also \emph{duel} two arms and get the arm with a greater mean. In our motivating application from crowdsourcing, dueling two arms can be more cost-effective and time-efficient than direct pulls. We refer to this problem as TBP with Dueling Choices (TBP-DC). This paper provides an algorithm called Rank-Search (RS) for solving TBP-DC by alternating between ranking and binary search. We prove theoretical g…
Google Research（谷歌研究院）；John Hopcroft Center（约翰·霍普克罗夫特中心）；Shanghai Jiao Tong University（上海交通大学）
Motivated by problems of learning to rank long item sequences, we introduce a variant of the cascading bandit model that considers flexible length sequences with varying rewards and losses. We formulate two generative models for this problem within the generalized linear setting, and design and analyze upper confidence algorithms for it. Our analysis delivers tight regret bounds which, when specialized to vanilla cascading bandits, results in sharper guarantees than previously available in the literature. We evaluate our algorithms on a number of real-world datasets, and show significantly imp…
Carnegie Mellon University（卡内基梅隆大学）；Google Research（谷歌研究院）；DeepMind；Duke University（杜克大学）；Norwegian University of Science and Technology（挪威科技大学）
We propose a novel framework for structured bandits, which we call an influence diagram bandit. Our framework captures complex statistical dependencies between actions, latent variables, and observations; and thus unifies and extends many existing models, such as combinatorial semi-bandits, cascading bandits, and low-rank bandits. We develop novel online learning algorithms that learn to act efficiently in our models. The key idea is to track a structured posterior distribution of model parameters, either exactly or approximately. To act, we sample model parameters from their posterior and the…
We introduce the probably approximately correct (PAC) \emph{Battling-Bandit} problem with the Plackett-Luce (PL) subset choice model--an online learning framework where at each trial the learner chooses a subset of $k$ arms from a fixed set of $n$ arms, and subsequently observes a stochastic feedback indicating preference information of the items in the chosen subset, e.g., the most preferred item or ranking of the top $m$ most preferred items etc. The objective is to identify a near-best item in the underlying PL model with high confidence. This generalizes the well-studied PAC \emph{Dueling-…
Testing is an important part of tackling the COVID-19 pandemic. Availability of testing is a bottleneck due to constrained resources and effective prioritization of individuals is necessary. Here, we discuss the impact of different prioritization policies on COVID-19 patient discovery and the ability of governments and health organizations to use the results for effective decision making. We suggest a framework for testing that balances the maximal discovery of positive individuals with the need for population-based surveillance aimed at understanding disease spread and characteristics. This f…
University of Amsterdam（阿姆斯特丹大学）；Ahold Delhaize（皇家阿霍德德尔海兹集团）；Microsoft（微软）
Ranking system is the core part of modern retrieval and recommender systems, where the goal is to rank candidate items given user contexts. Optimizing ranking systems online means that the deployed system can serve user requests, e.g., queries in the web search, and optimize the ranking policy by learning from user interactions, e.g., clicks. Bandit is a general online learning framework and can be used in our optimization task. However, due to the unique features of ranking, there are several challenges in designing bandit algorithms for ranking system optimization. In this dissertation, we s…
In many web applications, a recommendation is not a single item suggested to a user but a list of possibly interesting contents that may be ranked in some contexts. The combinatorial bandit problem has been studied quite extensively these last two years and many theoretical results now exist : lower bounds on the regret or asymptotically optimal algorithms. However, because of the variety of situations that can be considered, results are designed to solve the problem for a specific reward structure such as the Cascade Model. The present work focuses on the problem of ranking items when the use…
We consider Contextual Bandits with Concave Rewards (CBCR), a multi-objective bandit problem where the desired trade-off between the rewards is defined by a known concave objective function, and the reward vector depends on an observed stochastic context. We present the first algorithm with provably vanishing regret for CBCR without restrictions on the policy space, whereas prior works were restricted to finite policy spaces or tabular representations. Our solution is based on a geometric interpretation of CBCR algorithms as optimization algorithms over the convex set of expected rewards spann…
Dipartimento di Informatica Università degli Studi di Milano（米兰大学信息系）
Motivated by recommendation problems in music streaming platforms, we propose a nonstationary stochastic bandit model in which the expected reward of an arm depends on the number of rounds that have passed since the arm was last pulled. After proving that finding an optimal policy is NP-hard even when all model parameters are known, we introduce a class of ranking policies provably approximating, to within a constant factor, the expected reward of the optimal policy. We show an algorithm whose regret with respect to the best ranking policy is bounded by $\widetilde{\mathcal{O}}\big(\!\sqrt{kT}…
DEVCOM Army Research Laboratory（美国陆军研究实验室）；University of Maryland（马里兰大学）
In today's technology environment, information is abundant, dynamic, and heterogeneous in nature. Automated filtering and prioritization of information is based on the distinction between whether the information adds substantial value toward one's goal or not. Contextual multi-armed bandit has been widely used for learning to filter contents and prioritize according to user interest or relevance. Learn-to-Rank technique optimizes the relevance ranking on items, allowing the contents to be selected accordingly. We propose a novel approach to top-K rankings under the contextual multi-armed bandi…
CREST（法国经济与统计研究中心）；ENSAE Paris（巴黎国立统计与经济管理学院）；London School of Economics（伦敦政治经济学院）；Criteo AI Lab（Criteo人工智能实验室）
Finding an optimal matching in a weighted graph is a standard combinatorial problem. We consider its semi-bandit version where either a pair or a full matching is sampled sequentially. We prove that it is possible to leverage a rank-1 assumption on the adjacency matrix to reduce the sample complexity and the regret of off-the-shelf algorithms up to reaching a linear dependency in the number of vertices (up to poly log terms).
Indian Institute of Science（印度科学理工学院）；Inria Saclay – Île de France（法国国家信息与自动化研究所萨克雷-法兰西岛研究中心）
We study the task of maximizing rewards from recommending items (actions) to users sequentially interacting with a recommender system. Users are modeled as latent mixtures of C many representative user classes, where each class specifies a mean reward profile across actions. Both the user features (mixture distribution over classes) and the item features (mean reward vector per class) are unknown a priori. The user identity is the only contextual information available to the learner while interacting. This induces a low-rank structure on the matrix of expected rewards r a,b from recommending i…
University of California, Berkeley（加州大学伯克利分校）；Princeton University（普林斯顿大学）；University of Washington（华盛顿大学）
We give novel algorithms for multi-task and lifelong linear bandits with shared representation. Specifically, we consider the setting where we play $M$ linear bandits with dimension $d$, each for $T$ rounds, and these $M$ bandit tasks share a common $k(\ll d)$ dimensional linear representation. For both the multi-task setting where we play the tasks concurrently, and the lifelong setting where we play tasks sequentially, we come up with novel algorithms that achieve $\widetilde{O}\left(d\sqrt{kMT} + kM\sqrt{T}\right)$ regret bounds, which matches the known minimax regret lower bound up to loga…
Multi-armed bandit(MAB) problem is a reinforcement learning framework where an agent tries to maximise her profit by proper selection of actions through absolute feedback for each action. The dueling bandits problem is a variation of MAB problem in which an agent chooses a pair of actions and receives relative feedback for the chosen action pair. The dueling bandits problem is well suited for modelling a setting in which it is not possible to provide quantitative feedback for each action, but qualitative feedback for each action is preferred as in the case of human feedback. The dueling bandit…
Preventing poaching through ranger patrols protects endangered wildlife, directly contributing to the UN Sustainable Development Goal 15 of life on land. Combinatorial bandits have been used to allocate limited patrol resources, but existing approaches overlook the fact that each location is home to multiple species in varying proportions, so a patrol benefits each species to differing degrees. When some species are more vulnerable, we ought to offer more protection to these animals; unfortunately, existing combinatorial bandit approaches do not offer a way to prioritize important species. To …
