We consider a stochastic multi-armed bandit (MAB) problem with delayed impact of actions. In our setting, actions taken in the past impact the arm rewards in the subsequent future. This delayed impact of actions is prevalent in the real world. For example, the capability to pay back a loan for people in a certain social group might depend on historically how frequently that group has been approved loan applications. If banks keep rejecting loan applications to people in a disadvantaged group, it could create a feedback loop and further damage the chance of getting loans for people in that group. In this paper, we formulate this delayed and long-term impact of actions within the context of multi-armed bandits. We generalize the bandit setting to encode the dependency of this "bias" due to the action history during learning. The goal is to maximize the collected utilities over time while taking into account the dynamics created by the delayed impacts of historical actions. We propose an algorithm that achieves a regret of $\tilde{\mathcal{O}}(KT^{2/3})$ and show a matching regret lower bound of $Ω(KT^{2/3})$, where $K$ is the number of arms and $T$ is the learning horizon. Our results complement the bandit literature by adding techniques to deal with actions with long-term impacts and have implications in designing fair algorithms.
Lancaster University（兰卡斯特大学）；Columbia University（哥伦比亚大学）；DeepMind；University of Alberta（阿尔伯塔大学）
We study a variant of the stochastic $K$-armed bandit problem, which we call "bandits with delayed, aggregated anonymous feedback". In this problem, when the player pulls an arm, a reward is generated, however it is not immediately observed. Instead, at the end of each round the player observes only the sum of a number of previously generated rewards which happen to arrive in the given round. The rewards are stochastically delayed and due to the aggregated nature of the observations, the information of which arm led to a particular reward is lost. The question is what is the cost of the inform…
Stochastic linear bandits are a natural and well-studied model for structured exploration/exploitation problems and are widely used in applications such as online marketing and recommendation. One of the main challenges faced by practitioners hoping to apply existing algorithms is that usually the feedback is randomly delayed and delays are only partially observable. For example, while a purchase is usually observable some time after the display, the decision of not buying is never explicitly sent to the system. In other words, the learner only observes delayed positive events. We formalize th…
We explore a novel setting of the Multi-Armed Bandit (MAB) problem inspired from real world applications which we call bandits with "stochastic delayed composite anonymous feedback (SDCAF)". In SDCAF, the rewards on pulling arms are stochastic with respect to time but spread over a fixed number of time steps in the future after pulling the arm. The complexity of this problem stems from the anonymous feedback to the player and the stochastic generation of the reward. Due to the aggregated nature of the rewards, the player is unable to associate the reward to a particular time step from the past…
We propose Banker-OMD, a novel framework generalizing the classical Online Mirror Descent (OMD) technique in online learning algorithm design. Banker-OMD allows algorithms to robustly handle delayed feedback, and offers a general methodology for achieving $\tilde{O}(\sqrt{T} + \sqrt{D})$-style regret bounds in various delayed-feedback online learning tasks, where $T$ is the time horizon length and $D$ is the total feedback delay. We demonstrate the power of Banker-OMD with applications to three important bandit scenarios with delayed feedback, including delayed adversarial Multi-armed bandits …
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…
We study a multi-armed bandit problem with covariates in a setting where there is a possible delay in observing the rewards. Under some mild assumptions on the probability distributions for the delays and using an appropriate randomization to select the arms, the proposed strategy is shown to be strongly consistent.
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…
Stanford University（斯坦福大学）；University of Michigan（密歇根大学）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）
We propose a generalization of the best arm identification problem in stochastic multi-armed bandits (MAB) to the setting where every pull of an arm is associated with delayed feedback. The delay in feedback increases the effective sample complexity of standard algorithms, but can be offset if we have access to partial feedback received before a pull is completed. We propose a general framework to model the relationship between partial and delayed feedback, and as a special case we introduce efficient algorithms for settings where the partial feedback are biased or unbiased estimators of the d…
Delayed rewards problem in contextual bandits has been of interest in various practical settings. We study randomized allocation strategies and provide an understanding on how the exploration-exploitation tradeoff is affected by delays in observing the rewards. In randomized strategies, the extent of exploration-exploitation is controlled by a user-determined exploration probability sequence. In the presence of delayed rewards, one may choose between using the original exploration sequence that updates at every time point or update the sequence only when a new reward is observed, leading to tw…
University of Copenhagen（哥本哈根大学）；Google Research（谷歌研究院）
We present a modified tuning of the algorithm of Zimmert and Seldin [2020] for adversarial multiarmed bandits with delayed feedback, which in addition to the minimax optimal adversarial regret guarantee shown by Zimmert and Seldin simultaneously achieves a near-optimal regret guarantee in the stochastic setting with fixed delays. Specifically, the adversarial regret guarantee is $\mathcal{O}(\sqrt{TK} + \sqrt{dT\log K})$, where $T$ is the time horizon, $K$ is the number of arms, and $d$ is the fixed delay, whereas the stochastic regret guarantee is $\mathcal{O}\left(\sum_{i \neq i^*}(\frac{1}{…
MediaTek Research（联发科技研究院）；Imperial College London（帝国理工学院）
Black box optimisation of an unknown function from expensive and noisy evaluations is a ubiquitous problem in machine learning, academic research and industrial production. An abstraction of the problem can be formulated as a kernel based bandit problem (also known as Bayesian optimisation), where a learner aims at optimising a kernelized function through sequential noisy observations. The existing work predominantly assumes feedback is immediately available; an assumption which fails in many real world situations, including recommendation systems, clinical trials and hyperparameter tuning. We…
University of Southern California（南加州大学）；Tel Aviv University（特拉维夫大学）；Google Research（谷歌研究院）；Amazon（亚马逊）
The standard assumption in reinforcement learning (RL) is that agents observe feedback for their actions immediately. However, in practice feedback is often observed in delay. This paper studies online learning in episodic Markov decision process (MDP) with unknown transitions, adversarially changing costs, and unrestricted delayed bandit feedback. More precisely, the feedback for the agent in episode $k$ is revealed only in the end of episode $k + d^k$, where the delay $d^k$ can be changing over episodes and chosen by an oblivious adversary. We present the first algorithms that achieve near-o…
This paper deals with bandit online learning problems involving feedback of unknown delay that can emerge in multi-armed bandit (MAB) and bandit convex optimization (BCO) settings. MAB and BCO require only values of the objective function involved that become available through feedback, and are used to estimate the gradient appearing in the corresponding iterative algorithms. Since the challenging case of feedback with \emph{unknown} delays prevents one from constructing the sought gradient estimates, existing MAB and BCO algorithms become intractable. For such challenging setups, delayed expl…
We consider the problem where M agents collaboratively interact with an instance of a stochastic K-armed contextual bandit, where K>>M. The goal of the agents is to simultaneously minimize the cumulative regret over all the agents over a time horizon T. We consider a setting where the exact context is observed after a delay and at the time of choosing the action the agents are unaware of the context and only a distribution on the set of contexts is available. Such a situation arises in different applications where at the time of the decision the context needs to be predicted (e.g., weather for…
University of Pennsylvania（宾夕法尼亚大学）；Duke University（杜克大学）；Google（谷歌）
In this paper we initiate the study of optimization of bandit type problems in scenarios where the feedback of a play is not immediately known. This arises naturally in allocation problems which have been studied extensively in the literature, albeit in the absence of delays in the feedback. We study this problem in the Bayesian setting. In presence of delays, no solution with provable guarantees is known to exist with sub-exponential running time. We show that bandit problems with delayed feedback that arise in allocation settings can be forced to have significant structure, with a slight los…
In this paper, we present online algorithm called {\it Delaytron} for learning multi class classifiers using delayed bandit feedbacks. The sequence of feedback delays $\{d_t\}_{t=1}^T$ is unknown to the algorithm. At the $t$-th round, the algorithm observes an example $\mathbf{x}_t$ and predicts a label $\tilde{y}_t$ and receives the bandit feedback $\mathbb{I}[\tilde{y}_t=y_t]$ only $d_t$ rounds later. When $t+d_t>T$, we consider that the feedback for the $t$-th round is missing. We show that the proposed algorithm achieves regret of $\mathcal{O}\left(\sqrt{\frac{2 K}γ\left[\frac{T}{2}+\left(…
Delayed Bandits: When Do Intermediate Observations Help?
Università degli Studi di Milano（米兰大学）；Istituto Italiano di Tecnologia（意大利技术研究院）；University of Copenhagen（哥本哈根大学）；University of Amsterdam（阿姆斯特丹大学）；Politecnico di Milano（米兰理工大学）
Online advertising and product recommendation are important domains of applications for multi-armed bandit methods. In these fields, the reward that is immediately available is most often only a proxy for the actual outcome of interest, which we refer to as a conversion. For instance, in web advertising, clicks can be observed within a few seconds after an ad display but the corresponding sale --if any-- will take hours, if not days to happen. This paper proposes and investigates a new stochas-tic multi-armed bandit model in the framework proposed by Chapelle (2014) --based on empirical studie…
There is a rising interest in industrial online applications where data becomes available sequentially. Inspired by the recommendation of playlists to users where their preferences can be collected during the listening of the entire playlist, we study a novel bandit setting, namely Multi-Armed Bandit with Temporally-Partitioned Rewards (TP-MAB), in which the stochastic reward associated with the pull of an arm is partitioned over a finite number of consecutive rounds following the pull. This setting, unexplored so far to the best of our knowledge, is a natural extension of delayed-feedback ban…
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…
We study nonstochastic bandits and experts in a delayed setting where delays depend on both time and arms. While the setting in which delays only depend on time has been extensively studied, the arm-dependent delay setting better captures real-world applications at the cost of introducing new technical challenges. In the full information (experts) setting, we design an algorithm with a first-order regret bound that reveals an interesting trade-off between delays and losses. We prove a similar first-order regret bound also for the bandit setting, when the learner is allowed to observe how many …
University of Copenhagen（哥本哈根大学）；Università degli Studi di Milano（米兰大学）
We investigate multiarmed bandits with delayed feedback, where the delays need neither be identical nor bounded. We first prove that "delayed" Exp3 achieves the $O(\sqrt{(KT + D)\ln K} )$ regret bound conjectured by Cesa-Bianchi et al. [2019] in the case of variable, but bounded delays. Here, $K$ is the number of actions and $D$ is the total delay over $T$ rounds. We then introduce a new algorithm that lifts the requirement of bounded delays by using a wrapper that skips rounds with excessively large delays. The new algorithm maintains the same regret bound, but similar to its predecessor requ…
Media services providers, such as music streaming platforms, frequently leverage swipeable carousels to recommend personalized content to their users. However, selecting the most relevant items (albums, artists, playlists...) to display in these carousels is a challenging task, as items are numerous and as users have different preferences. In this paper, we model carousel personalization as a contextual multi-armed bandit problem with multiple plays, cascade-based updates and delayed batch feedback. We empirically show the effectiveness of our framework at capturing characteristics of real-wor…
Otto-von-Guericke University of Magdeburg（马格德堡奥托·冯·格里克大学）；GoogleDeepMind（谷歌深度思维）
Significant work has been recently dedicated to the stochastic delayed bandit setting because of its relevance in applications. The applicability of existing algorithms is however restricted by the fact that strong assumptions are often made on the delay distributions, such as full observability, restrictive shape constraints, or uniformity over arms. In this work, we weaken them significantly and only assume that there is a bound on the tail of the delay. In particular, we cover the important case where the delay distributions vary across arms, and the case where the delays are heavy-tailed. …
Stochastic delays in feedback lead to unstable sequential learning using multi-armed bandits. Recently, empirical Bayesian shrinkage has been shown to improve reward estimation in bandit learning. Here, we propose a novel adaptation to shrinkage that estimates smoothed reward estimates from windowed cumulative inputs, to deal with incomplete knowledge from delayed feedback and non-stationary rewards. Using numerical simulations, we show that this adaptation retains the benefits of shrinkage, and improves the stability of reward estimation by more than 50%. Our proposal reduces variability in t…
Eindhoven University of Technology（埃因霍温理工大学）；Jheronimus Academy of Data Science（赫罗尼姆斯数据科学学院）；University in Jyväskylä（于韦斯屈莱大学）；Tilburg University（蒂尔堡大学）
We consider a special case of bandit problems, namely batched bandits. Motivated by natural restrictions of recommender systems and e-commerce platforms, we assume that a learning agent observes responses batched in groups over a certain time period. Unlike previous work, we consider a more practically relevant batch-centric scenario of batch learning. We provide a policy-agnostic regret analysis and demonstrate upper and lower bounds for the regret of a candidate policy. Our main theoretical results show that the impact of batch learning can be measured in terms of online behavior. Finally, w…
Adversarial Bandits with Multi-User Delayed Feedback: Theory and Application
Hong Kong Baptist University（香港浸会大学）；BNU-HKBU United International College（北京师范大学-香港浸会大学联合国际学院）；Advanced Institute of Natural Sciences, Beijing Normal University（北京师范大学先进自然科学研究院）
This paper studies the problem of stochastic continuum-armed bandit with constraints (SCBwC), where we optimize a black-box reward function $f(x)$ subject to a black-box constraint function $g(x)\leq 0$ over a continuous space $\mathcal X$. We model reward and constraint functions via Gaussian processes (GPs) and propose a Rectified Pessimistic-Optimistic Learning framework (RPOL), a penalty-based method incorporating optimistic and pessimistic GP bandit learning for reward and constraint functions, respectively. We consider the metric of cumulative constraint violation $\sum_{t=1}^T(g(x_t))^{…
We consider the Scale-Free Adversarial Multi-Armed Bandit (MAB) problem with unrestricted feedback delays. In contrast to the standard assumption that all losses are $[0,1]$-bounded, in our setting, losses can fall in a general bounded interval $[-L, L]$, unknown to the agent beforehand. Furthermore, the feedback of each arm pull can experience arbitrary delays. We propose a novel approach named Scale-Free Delayed INF (SFD-INF) for this novel setting, which combines a recent "convex combination trick" together with a novel doubling and skipping technique. We then present two instances of SFD-I…
Decision-making problems of sequential nature, where decisions made in the past may have an impact on the future, are used to model many practically important applications. In some real-world applications, feedback about a decision is delayed and may arrive via partial rewards that are observed with different delays. Motivated by such scenarios, we propose a novel problem formulation called multi-armed bandits with generalized temporally-partitioned rewards. To formalize how feedback about a decision is partitioned across several time steps, we introduce $β$-spread property. We derive a lower …
Regret Bounds for Adversarial Contextual Bandits with General Function Approximation and Delayed Feedback
Blavatnik School of Computer Science Tel Aviv University（塔尔医学院计算机科学学院）；School of Electrical Engineering Tel Aviv University（塔尔医学院电气工程学院）；Google Research（谷歌研究院）
We determine sharp bounds on the price of bandit feedback for several variants of the mistake-bound model. The first part of the paper presents bounds on the $r$-input weak reinforcement model and the $r$-input delayed, ambiguous reinforcement model. In both models, the adversary gives $r$ inputs in each round and only indicates a correct answer if all $r$ guesses are correct. The only difference between the two models is that in the delayed, ambiguous model, the learner must answer each input before receiving the next input of the round, while the learner receives all $r$ inputs at once in th…
Università degli Studi di Milano（米兰大学）；Paul Sabatier University (UT3)（保罗·萨巴蒂埃大学（UT3））；Toulouse School of Economics (TSE)（图卢兹经济学院（TSE））；Istituto Italiano di Tecnologia (IIT)（意大利技术研究院（IIT））；Google Research（谷歌研究院）；Tel Aviv University（特拉维夫大学）
We investigate a nonstochastic bandit setting in which the loss of an action is not immediately charged to the player, but rather spread over the subsequent rounds in an adversarial way. The instantaneous loss observed by the player at the end of each round is then a sum of many loss components of previously played actions. This setting encompasses as a special case the easier task of bandits with delayed feedback, a well-studied framework where the player observes the delayed losses individually. Our first contribution is a general reduction transforming a standard bandit algorithm into one t…
Princeton University（普林斯顿大学）；Massachusetts Institute of Technology（麻省理工学院）
The cooperative bandit problem is increasingly becoming relevant due to its applications in large-scale decision-making. However, most research for this problem focuses exclusively on the setting with perfect communication, whereas in most real-world distributed settings, communication is often over stochastic networks, with arbitrary corruptions and delays. In this paper, we study cooperative bandit learning under three typical real-world communication scenarios, namely, (a) message-passing over stochastic time-varying networks, (b) instantaneous reward-sharing over a network with random dela…
A Reduction-based Framework for Sequential Decision Making with Delayed Feedback
Center for Data Science, Peking University（北京大学数据科学中心）；Zhejiang Lab（之江实验室）；National Key Laboratory of General Artificial Intelligence（通用人工智能全国重点实验室）；School of Intelligence Science and Technology, Peking University（北京大学智能科学与技术学院）；University of Washington（华盛顿大学）；University of California, Berkeley（加州大学伯克利分校）
