The statistical framework of Generalized Linear Models (GLM) can be applied to sequential problems involving categorical or ordinal rewards associated, for instance, with clicks, likes or ratings. In the example of binary rewards, logistic regression is well-known to be preferable to the use of standard linear modeling. Previous works have shown how to deal with GLMs in contextual online learning with bandit feedback when the environment is assumed to be stationary. In this paper, we relax this latter assumption and propose two upper confidence bound based algorithms that make use of either a sliding window or a discounted maximum-likelihood estimator. We provide theoretical guarantees on the behavior of these algorithms for general context sequences and in the presence of abrupt changes. These results take the form of high probability upper bounds for the dynamic regret that are of order d^2/3 G^1/3 T^2/3 , where d, T and G are respectively the dimension of the unknown parameter, the number of rounds and the number of breakpoints up to time T. The empirical performance of the algorithms is illustrated in simulated environments.
Non-stationary bandits and online clustering of bandits lift the restrictive assumptions in contextual bandits and provide solutions to many important real-world scenarios. Though the essence in solving these two problems overlaps considerably, they have been studied independently. In this paper, we connect these two strands of bandit research under the notion of test of homogeneity, which seamlessly addresses change detection for non-stationary bandit and cluster identification for online clustering of bandit in a unified solution framework. Rigorous regret analysis and extensive empirical ev…
University of California, Davis（加利福尼亚大学戴维斯分校）；University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
Classic contextual bandit algorithms for linear models, such as LinUCB, assume that the reward distribution for an arm is modeled by a stationary linear regression. When the linear regression model is non-stationary over time, the regret of LinUCB can scale linearly with time. In this paper, we propose a novel multiscale changepoint detection method for the non-stationary linear bandit problems, called Multiscale-LinUCB, which actively adapts to the changing environment. We also provide theoretical analysis of regret bound for Multiscale-LinUCB algorithm. Experimental results show that our pro…
We propose an algorithm for non-stationary kernel bandits that does not require prior knowledge of the degree of non-stationarity. The algorithm follows randomized strategies obtained by solving optimization problems that balance exploration and exploitation. It adapts to non-stationarity by restarting when a change in the reward function is detected. Our algorithm enjoys a tighter dynamic regret bound than previous work on the non-stationary kernel bandit setting. Moreover, when applied to the non-stationary linear bandit setting by using a linear kernel, our algorithm is nearly minimax optim…
We investigate the piecewise-stationary combinatorial semi-bandit problem. Compared to the original combinatorial semi-bandit problem, our setting assumes the reward distributions of base arms may change in a piecewise-stationary manner at unknown time steps. We propose an algorithm, \texttt{GLR-CUCB}, which incorporates an efficient combinatorial semi-bandit algorithm, \texttt{CUCB}, with an almost parameter-free change-point detector, the \emph{Generalized Likelihood Ratio Test} (GLRT). Our analysis shows that the regret of \texttt{GLR-CUCB} is upper bounded by $\mathcal{O}(\sqrt{NKT\log{T}}…
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…
Paderborn University（帕德博恩大学）；University of Munich（慕尼黑大学）
We study the non-stationary dueling bandits problem with $K$ arms, where the time horizon $T$ consists of $M$ stationary segments, each of which is associated with its own preference matrix. The learner repeatedly selects a pair of arms and observes a binary preference between them as feedback. To minimize the accumulated regret, the learner needs to pick the Condorcet winner of each stationary segment as often as possible, despite preference matrices and segment lengths being unknown. We propose the $\mathrm{Beat\, the\, Winner\, Reset}$ algorithm and prove a bound on its expected binary weak…
Indian Institute of Technology, Madras（印度理工学院马德拉斯分校）
We consider the multi armed bandit problem in non-stationary environments. Based on the Bayesian method, we propose a variant of Thompson Sampling which can be used in both rested and restless bandit scenarios. Applying discounting to the parameters of prior distribution, we describe a way to systematically reduce the effect of past observations. Further, we derive the exact expression for the probability of picking sub-optimal arms. By increasing the exploitative value of Bayes' samples, we also provide an optimistic version of the algorithm. Extensive empirical analysis is conducted under va…
We consider minimisation of dynamic regret in non-stationary bandits with a slowly varying property. Namely, we assume that arms' rewards are stochastic and independent over time, but that the absolute difference between the expected rewards of any arm at any two consecutive time-steps is at most a drift limit $δ> 0$. For this setting that has not received enough attention in the past, we give a new algorithm which extends naturally the well-known Successive Elimination algorithm to the non-stationary bandit setting. We establish the first instance-dependent regret upper bound for slowly varyi…
Naveen Jindal School of Management, University of Texas at Dallas（德克萨斯大学达拉斯分校纳文·金达尔管理学院）
In this paper we study the non-stationary stochastic optimization question with bandit feedback and dynamic regret measures. The seminal work of Besbes et al. (2015) shows that, when aggregated function changes is known a priori, a simple re-starting algorithm attains the optimal dynamic regret. In this work, we designed a stochastic optimization algorithm with fixed step sizes, which combined together with the multi-scale sampling framework of Wei and Luo (2021) achieves the optimal dynamic regret in non-stationary stochastic optimization without requiring prior knowledge of function change b…
Non-Stationary Bandit Learning via Predictive Sampling
Jones Graduate School of Business, Rice University（里士满大学沃森商学院研究生院）；Stanford Graduate School of Business（斯坦福商学院）；Department of Management Science and Engineering, Department of Electrical Engineering, Stanford University（斯坦福大学管理科学与工程系、电气工程系）
Online recommender systems often face long delays in receiving feedback, especially when optimizing for some long-term metrics. While mitigating the effects of delays in learning is well-understood in stationary environments, the problem becomes much more challenging when the environment changes. In fact, if the timescale of the change is comparable to the delay, it is impossible to learn about the environment, since the available observations are already obsolete. However, the arising issues can be addressed if intermediate signals are available without delay, such that given those signals, t…
University of Southern California（南加州大学）；Microsoft Research, NYC（微软研究院（纽约））
Most contextual bandit algorithms minimize regret against the best fixed policy, a questionable benchmark for non-stationary environments that are ubiquitous in applications. In this work, we develop several efficient contextual bandit algorithms for non-stationary environments by equipping existing methods for i.i.d. problems with sophisticated statistical tests so as to dynamically adapt to a change in distribution. We analyze various standard notions of regret suited to non-stationary environments for these algorithms, including interval regret, switching regret, and dynamic regret. When co…
University of Oslo（奥斯陆大学）；Toyota Technological Institute at Chicago（芝加哥丰田技术研究所）
We study the problem of non-stationary dueling bandits and provide the first adaptive dynamic regret algorithm for this problem. The only two existing attempts in this line of work fall short across multiple dimensions, including pessimistic measures of non-stationary complexity and non-adaptive parameter tuning that requires knowledge of the number of preference changes. We develop an elimination-based rescheduling algorithm to overcome these shortcomings and show a near-optimal $\tilde{O}(\sqrt{S^{\texttt{CW}} T})$ dynamic regret bound, where $S^{\texttt{CW}}$ is the number of times the Cond…
Multi-armed bandit algorithms have become a reference solution for handling the explore/exploit dilemma in recommender systems, and many other important real-world problems, such as display advertisement. However, such algorithms usually assume a stationary reward distribution, which hardly holds in practice as users' preferences are dynamic. This inevitably costs a recommender system consistent suboptimal performance. In this paper, we consider the situation where the underlying distribution of reward remains unchanged over (possibly short) epochs and shifts at unknown time instants. In accor…
Dynamic Regret for Non-Stationary Linear Bandits via Misspecification Reductions
Department of Mathematics, The Hong Kong University of Science and Technology（香港科技大学数学系）；Stern School of Business, New York University（纽约大学斯特恩商学院）；Department of IEDA, The Hong Kong University of Science and Technology（香港科技大学工业工程与决策分析系）
We study the non-stationary stochastic multi-armed bandit problem, where the reward statistics of each arm may change several times during the course of learning. The performance of a learning algorithm is evaluated in terms of their dynamic regret, which is defined as the difference between the expected cumulative reward of an agent choosing the optimal arm in every time step and the cumulative reward of the learning algorithm. One way to measure the hardness of such environments is to consider how many times the identity of the optimal arm can change. We propose a method that achieves, in $K…
Multi-armed bandit (MAB) is a class of online learning problems where a learning agent aims to maximize its expected cumulative reward while repeatedly selecting to pull arms with unknown reward distributions. We consider a scenario where the reward distributions may change in a piecewise-stationary fashion at unknown time steps. We show that by incorporating a simple change-detection component with classic UCB algorithms to detect and adapt to changes, our so-called M-UCB algorithm can achieve nearly optimal regret bound on the order of $O(\sqrt{MKT\log T})$, where $T$ is the number of time s…
We propose the first contextual bandit algorithm that is parameter-free, efficient, and optimal in terms of dynamic regret. Specifically, our algorithm achieves dynamic regret $\mathcal{O}(\min\{\sqrt{ST}, Δ^{\frac{1}{3}}T^{\frac{2}{3}}\})$ for a contextual bandit problem with $T$ rounds, $S$ switches and $Δ$ total variation in data distributions. Importantly, our algorithm is adaptive and does not need to know $S$ or $Δ$ ahead of time, and can be implemented efficiently assuming access to an ERM oracle. Our results strictly improve the $\mathcal{O}(\min \{S^{\frac{1}{4}}T^{\frac{3}{4}}, Δ^{\f…
Massachusetts Institute of Technology（麻省理工学院）；Harvard University（哈佛大学）
We consider the problem of designing an allocation rule or an "online learning algorithm" for a class of bandit problems in which the set of control actions available at each time $s$ is a convex, compact subset of $\mathbb{R}^d$. Upon choosing an action $x$ at time $s$, the algorithm obtains a noisy value of the unknown and time-varying function $f_s$ evaluated at $x$. The "regret" of an algorithm is the gap between its expected reward, and the reward earned by a strategy which has the knowledge of the function $f_s$ at each time $s$ and hence chooses the action $x_s$ that maximizes $f_s$. Fo…
Centre Borelli, ENS Paris-Saclay（巴黎萨克雷高等师范学院博雷利中心）；Cdiscount（Cdiscount公司）；CREST, ENSAE Paris（巴黎经济与统计高等学校统计与经济研究中心）；Criteo AI Lab（Criteo人工智能实验室）
Continuously learning and leveraging the knowledge accumulated from prior tasks in order to improve future performance is a long standing machine learning problem. In this paper, we study the problem in the multi-armed bandit framework with the objective to minimize the total regret incurred over a series of tasks. While most bandit algorithms are designed to have a low worst-case regret, we examine here the average regret over bandit instances drawn from some prior distribution which may change over time. We specifically focus on confidence interval tuning of UCB algorithms. We propose a band…
We consider a stochastic linear bandit model in which the available actions correspond to arbitrary context vectors whose associated rewards follow a non-stationary linear regression model. In this setting, the unknown regression parameter is allowed to vary in time. To address this problem, we propose D-LinUCB, a novel optimistic algorithm based on discounted linear regression, where exponential weights are used to smoothly forget the past. This involves studying the deviations of the sequential weighted least-squares estimator under generic assumptions. As a by-product, we obtain novel devia…
The multi-armed bandit problem has been extensively studied under the stationary assumption. However in reality, this assumption often does not hold because the distributions of rewards themselves may change over time. In this paper, we propose a change-detection (CD) based framework for multi-armed bandit problems under the piecewise-stationary setting, and study a class of change-detection based UCB (Upper Confidence Bound) policies, CD-UCB, that actively detects change points and restarts the UCB indices. We then develop CUSUM-UCB and PHT-UCB, that belong to the CD-UCB class and use cumulat…
Motivated by drug design, we consider the best-arm identification problem in generalized linear bandits. More specifically, we assume each arm has a vector of covariates, there is an unknown vector of parameters that is common across the arms, and a generalized linear model captures the dependence of rewards on the covariate and parameter vectors. The problem is to minimize the number of arm pulls required to identify an arm that is sufficiently close to optimal with a sufficiently high probability. Building on recent progress in best-arm identification for linear bandits (Xu et al. 2018), we …
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；University of Minnesota - Twin Cities（明尼苏达大学双城分校）
Cascading bandit (CB) is a popular model for web search and online advertising, where an agent aims to learn the $K$ most attractive items out of a ground set of size $L$ during the interaction with a user. However, the stationary CB model may be too simple to apply to real-world problems, where user preferences may change over time. Considering piecewise-stationary environments, two efficient algorithms, \texttt{GLRT-CascadeUCB} and \texttt{GLRT-CascadeKL-UCB}, are developed and shown to ensure regret upper bounds on the order of $\mathcal{O}(\sqrt{NLT\log{T}})$, where $N$ is the number of pi…
We study the multi-armed bandit problem where the rewards are realizations of general non-stationary stochastic processes, a setting that generalizes many existing lines of work and analyses. In particular, we present a theoretical analysis and derive regret guarantees for rested bandits in which the reward distribution of each arm changes only when we pull that arm. Remarkably, our regret bounds are logarithmic in the number of rounds under several natural conditions. We introduce a new algorithm based on classical UCB ideas combined with the notion of weighted discrepancy, a useful tool for …
This paper tackles the risk averse multi-armed bandits problem when incurred losses are non-stationary. The conditional value-at-risk (CVaR) is used as the objective function. Two estimation methods are proposed for this objective function in the presence of non-stationary losses, one relying on a weighted empirical distribution of losses and another on the dual representation of the CVaR. Such estimates can then be embedded into classic arm selection methods such as epsilon-greedy policies. Simulation experiments assess the performance of the arm selection algorithms based on the two novel es…
Institute of Science and Technology Austria（奥地利科学技术研究所）
We consider the problem of minimizing the regret in stochastic multi-armed bandit, when the measure of goodness of an arm is not the mean return, but some general function of the mean and the variance.We characterize the conditions under which learning is possible and present examples for which no natural algorithm can achieve sublinear regret.
A survey is performed of various Multi-Armed Bandit (MAB) strategies in order to examine their performance in circumstances exhibiting non-stationary stochastic reward functions in conjunction with delayed feedback. We run several MAB simulations to simulate an online eCommerce platform for grocery pick up, optimizing for product availability. In this work, we evaluate several popular MAB strategies, such as $ε$-greedy, UCB1, and Thompson Sampling. We compare the respective performances of each MAB strategy in the context of regret minimization. We run the analysis in the scenario where the re…
We study the bandit problem where arms are associated with stationary phi-mixing processes and where rewards are therefore dependent: the question that arises from this setting is that of recovering some independence by ignoring the value of some rewards. As we shall see, the bandit problem we tackle requires us to address the exploration/exploitation/independence trade-off. To do so, we provide a UCB strategy together with a general regret analysis for the case where the size of the independence blocks (the ignored rewards) is fixed and we go a step beyond by providing an algorithm that is ab…
We consider the non-stationary multi-armed bandit (MAB) framework and propose a Kolmogorov-Smirnov (KS) test based Thompson Sampling (TS) algorithm named TS-KS, that actively detects change points and resets the TS parameters once a change is detected. In particular, for the two-armed bandit case, we derive bounds on the number of samples of the reward distribution to detect the change once it occurs. Consequently, we show that the proposed algorithm has sub-linear regret. Contrary to existing works, our algorithm is able to detect a change when the underlying reward distribution changes even …
We investigate two perturbation approaches to overcome conservatism that optimism based algorithms chronically suffer from in practice. The first approach replaces optimism with a simple randomization when using confidence sets. The second one adds random perturbations to its current estimate before maximizing the expected reward. For non-stationary linear bandits, where each action is associated with a $d$-dimensional feature and the unknown parameter is time-varying with total variation $B_T$, we propose two randomized algorithms, Discounted Randomized LinUCB (D-RandLinUCB) and Discounted Li…
Users of recommender systems often behave in a non-stationary fashion, due to their evolving preferences and tastes over time. In this work, we propose a practical approach for fast personalization to non-stationary users. The key idea is to frame this problem as a latent bandit, where the prototypical models of user behavior are learned offline and the latent state of the user is inferred online from its interactions with the models. We call this problem a non-stationary latent bandit. We propose Thompson sampling algorithms for regret minimization in non-stationary latent bandits, analyze th…
We consider a Multi-Armed Bandit problem in which the rewards are non-stationary and are dependent on past actions and potentially on past contexts. At the heart of our method, we employ a recurrent neural network, which models these sequences. In order to balance between exploration and exploitation, we present an energy minimization term that prevents the neural network from becoming too confident in support of a certain action. This term provably limits the gap between the maximal and minimal probabilities assigned by the network. In a diverse set of experiments, we demonstrate that our met…
We introduce GLR-klUCB, a novel algorithm for the piecewise iid non-stationary bandit problem with bounded rewards. This algorithm combines an efficient bandit algorithm, kl-UCB, with an efficient, parameter-free, changepoint detector, the Bernoulli Generalized Likelihood Ratio Test, for which we provide new theoretical guarantees of independent interest. Unlike previous non-stationary bandit algorithms using a change-point detector, GLR-klUCB does not need to be calibrated based on prior knowledge on the arms' means. We prove that this algorithm can attain a $O(\sqrt{TA Υ_T\log(T)})$ regret i…
The Multi-Armed Bandits (MAB) framework highlights the tension between acquiring new knowledge (Exploration) and leveraging available knowledge (Exploitation). In the classical MAB problem, a decision maker must choose an arm at each time step, upon which she receives a reward. The decision maker's objective is to maximize her cumulative expected reward over the time horizon. The MAB problem has been studied extensively, specifically under the assumption of the arms' rewards distributions being stationary, or quasi-stationary, over time. We consider a variant of the MAB framework, which we ter…
Otto-von-Guericke University of Magdeburg（马格德堡奥托·冯·格里克大学）；University of Warwick（华威大学）
In this paper, we study a non-stationary stochastic bandit problem, which generalizes the switching bandit problem. On top of the switching bandit problem (\textbf{Case a}), we are interested in three concrete examples: (\textbf{b}) the means of the arms are local polynomials, (\textbf{c}) the means of the arms are locally smooth, and (\textbf{d}) the gaps of the arms have a bounded number of inflexion points and where the highest arm mean cannot vary too much in a short range. These three settings are very different, but have in common the following: (i) the number of similarly-sized level se…
Stanford University（斯坦福大学）；University of Oxford（牛津大学）；New York University（纽约大学）
In this paper, we consider online learning in generalized linear contextual bandits where rewards are not immediately observed. Instead, rewards are available to the decision-maker only after some delay, which is unknown and stochastic. We study the performance of two well-known algorithms adapted to this delayed setting: one based on upper confidence bounds, and the other based on Thompson sampling. We describe modifications on how these two algorithms should be adapted to handle delays and give regret characterizations for both algorithms. Our results contribute to the broad landscape of con…
University of Edinburgh（爱丁堡大学）；University College London（伦敦大学学院）；Nara Institute of Science and Technology（奈良先端科学技术大学院大学）
Active learning aims to reduce annotation cost by predicting which samples are useful for a human teacher to label. However it has become clear there is no best active learning algorithm. Inspired by various philosophies about what constitutes a good criteria, different algorithms perform well on different datasets. This has motivated research into ensembles of active learners that learn what constitutes a good criteria in a given scenario, typically via multi-armed bandit algorithms. Though algorithm ensembles can lead to better results, they overlook the fact that not only does algorithm eff…
Single Index Bandits: Generalized Linear Contextual Bandits with Unknown Reward Functions
Microsoft（微软公司）；University of California, Davis（加州大学戴维斯分校）；University of North Carolina at Chapel Hill（北卡罗来纳大学教堂山分校）；University of California, Los Angeles（加州大学洛杉矶分校）；National University of Singapore（新加坡国立大学）
