Online learning with delayed feedback has received increasing attention recently due to its several applications in distributed, web-based learning problems. In this paper we provide a systematic study of the topic, and analyze the effect of delay on the regret of online learning algorithms. Somewhat surprisingly, it turns out that delay increases the regret in a multiplicative way in adversarial problems, and in an additive way in stochastic problems. We give meta-algorithms that transform, in a black-box fashion, algorithms developed for the non-delayed case into ones that can handle the presence of delays in the feedback loop. Modifications of the well-known UCB algorithm are also developed for the bandit problem with delayed feedback, with the advantage over the meta-algorithms that they can be implemented with lower complexity.
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…
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 …
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…
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 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…
Tel Aviv University（特拉维夫大学）；Google Research（谷歌研究院）
Reinforcement learning typically assumes that agents observe feedback for their actions immediately, but in many real-world applications (like recommendation systems) feedback is observed in delay. This paper studies online learning in episodic Markov decision processes (MDPs) with unknown transitions, adversarially changing costs and unrestricted delayed feedback. That is, the costs and trajectory of episode $k$ are revealed to the learner only in the end of episode $k + d^k$, where the delays $d^k$ are neither identical nor bounded, and are chosen by an oblivious adversary. We present novel …
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…
Conducting experiments with objectives that take significant delays to materialize (e.g. conversions, add-to-cart events, etc.) is challenging. Although the classical "split sample testing" is still valid for the delayed feedback, the experiment will take longer to complete, which also means spending more resources on worse-performing strategies due to their fixed allocation schedules. Alternatively, adaptive approaches such as "multi-armed bandits" are able to effectively reduce the cost of experimentation. But these methods generally cannot handle delayed objectives directly out of the box. …
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…
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）
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…
Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究院）
The article is devoted to investigating the application of hedging strategies to online expert weight allocation under delayed feedback. As the main result, we develop the General Hedging algorithm $\mathcal{G}$ based on the exponential reweighing of experts' losses. We build the artificial probabilistic framework and use it to prove the adversarial loss bounds for the algorithm $\mathcal{G}$ in the delayed feedback setting. The designed algorithm $\mathcal{G}$ can be applied to both countable and continuous sets of experts. We also show how algorithm $\mathcal{G}$ extends classical Hedge (Mul…
Bayesian optimization (BO) is a widely-used sequential method for zeroth-order optimization of complex and expensive-to-compute black-box functions. The existing BO methods assume that the function evaluation (feedback) is available to the learner immediately or after a fixed delay. Such assumptions may not be practical in many real-life problems like online recommendations, clinical trials, and hyperparameter tuning where feedback is available after a random delay. To benefit from the experimental parallelization in these problems, the learner needs to start new function evaluations without w…
In display advertising, predicting the conversion rate, that is, the probability that a user takes a predefined action on an advertiser's website, such as purchasing goods is fundamental in estimating the value of displaying the advertisement. However, there is a relatively long time delay between a click and its resultant conversion. Because of the delayed feedback, some positive instances at the training period are labeled as negative because some conversions have not yet occurred when training data are gathered. As a result, the conditional label distributions differ between the training da…
Capacity-Constrained Online Convex Optimization with Delayed Feedback
Department of Statistical Sciences, University of Toronto（多伦多大学统计科学系）；Vector Institute（向量研究所）；Institute for Data, Econometrics, Algorithms, and Learning (IDEAL), hosted by UIC and TTIC（数据、计量经济学、算法与学习研究所（IDEAL），由伊利诺伊大学芝加哥分校和丰田工业大学芝加哥分校主办）
We consider a system where agents enter in an online fashion and are evaluated based on their attributes or context vectors. There can be practical situations where this context is partially observed, and the unobserved part comes after some delay. We assume that an agent, once left, cannot re-enter the system. Therefore, the job of the system is to provide an estimated score for the agent based on her instantaneous score and possibly some inference of the instantaneous score over the delayed score. In this paper, we estimate the delayed context via an online convex game between the agent and …
In black-box optimization problems, we aim to maximize an unknown objective function, where the function is only accessible through feedbacks of an evaluation or simulation oracle. In real-life, the feedbacks of such oracles are often noisy and available after some unknown delay that may depend on the computation time of the oracle. Additionally, if the exact evaluations are expensive but coarse approximations are available at a lower cost, the feedbacks can have multi-fidelity. In order to address this problem, we propose a generic extension of hierarchical optimistic tree search (HOO), calle…
The Alan Turing Institute（艾伦·图灵研究所）；WMG Cyber Security Centre, University of Warwick（华威大学WMG网络安全中心）
The increasing connectivity of data and cyber-physical systems has resulted in a growing number of cyber-attacks. Real-time detection of such attacks, through the identification of anomalous activity, is required so that mitigation and contingent actions can be effectively and rapidly deployed. We propose a new approach for aggregating unsupervised anomaly detection algorithms and incorporating feedback when it becomes available. We apply this approach to open-source real datasets and show that both aggregating models, which we call experts, and incorporating feedback significantly improve the…
University of California, Berkeley（加州大学伯克利分校）；Microsoft Research（微软研究院）
Motivated by applications to online learning in sparse estimation and Bayesian optimization, we consider the problem of online unconstrained nonsubmodular minimization with delayed costs in both full information and bandit feedback settings. In contrast to previous works on online unconstrained submodular minimization, we focus on a class of nonsubmodular functions with special structure, and prove regret guarantees for several variants of the online and approximate online bandit gradient descent algorithms in static and delayed scenarios. We derive bounds for the agent's regret in the full in…
University of Chinese Academy of Sciences（中国科学院大学）；Tencent（腾讯）
The delayed feedback problem is one of the imperative challenges in online advertising, which is caused by the highly diversified feedback delay of a conversion varying from a few minutes to several days. It is hard to design an appropriate online learning system under these non-identical delay for different types of ads and users. In this paper, we propose to tackle the delayed feedback problem in online advertising by "Following the Prophet" (FTP for short). The key insight is that, if the feedback came instantly for all the logged samples, we could get a model without delayed feedback, name…
Alleviating the delayed feedback problem is of crucial importance for the conversion rate(CVR) prediction in online advertising. Previous delayed feedback modeling methods using an observation window to balance the trade-off between waiting for accurate labels and consuming fresh feedback. Moreover, to estimate CVR upon the freshly observed but biased distribution with fake negatives, the importance sampling is widely used to reduce the distribution bias. While effective, we argue that previous approaches falsely treat fake negative samples as real negative during the importance weighting and …
Conversion rate (CVR) prediction is one of the most critical tasks for digital display advertising. Commercial systems often require to update models in an online learning manner to catch up with the evolving data distribution. However, conversions usually do not happen immediately after a user click. This may result in inaccurate labeling, which is called delayed feedback problem. In previous studies, delayed feedback problem is handled either by waiting positive label for a long period of time, or by consuming the negative sample on its arrival and then insert a positive duplicate when a con…
