Conservative mechanism is a desirable property in decision-making problems which balance the tradeoff between the exploration and exploitation. We propose the novel \emph{conservative contextual combinatorial cascading bandit ($C^4$-bandit)}, a cascading online learning game which incorporates the conservative mechanism. At each time step, the learning agent is given some contexts and has to recommend a list of items but not worse than the base strategy and then observes the reward by some stopping rules. We design the $C^4$-UCB algorithm to solve the problem and prove its n-step upper regret bound for two situations: known baseline reward and unknown baseline reward. The regret in both situations can be decomposed into two terms: (a) the upper bound for the general contextual combinatorial cascading bandit; and (b) a constant term for the regret from the conservative mechanism. We also improve the bound of the conservative contextual combinatorial bandit as a by-product. Experiments on synthetic data demonstrate its advantages and validate our theoretical analysis.
Motivated by the pressing need for efficient optimization in online recommender systems, we revisit the cascading bandit model proposed by Kveton et al. (2015). While Thompson sampling (TS) algorithms have been shown to be empirically superior to Upper Confidence Bound (UCB) algorithms for cascading bandits, theoretical guarantees are only known for the latter. In this paper, we first provide a problem-dependent upper bound on the regret of a TS algorithm with Beta-Bernoulli updates; this upper bound is tighter than a recent derivation under a more general setting by Huyuk and Tekin (2019). Ne…
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…
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…
UT Austin（德克萨斯大学奥斯汀分校）；UIUC（伊利诺伊大学厄巴纳-香槟分校）；Amazon（亚马逊）
Cascading bandits is a natural and popular model that frames the task of learning to rank from Bernoulli click feedback in a bandit setting. For the case of unstructured rewards, we prove matching upper and lower bounds for the problem-independent (i.e., gap-free) regret, both of which strictly improve the best known. A key observation is that the hard instances of this problem are those with small mean rewards, i.e., the small click-through rates that are most relevant in practice. Based on this, and the fact that small mean implies small variance for Bernoullis, our key technical result show…
Adobe Research（奥多比研究院）；Yahoo Labs（雅虎实验室）；Technicolor Research（特艺集团研究院）；University of Alberta（阿尔伯塔大学）
We propose combinatorial cascading bandits, a class of partial monitoring problems where at each step a learning agent chooses a tuple of ground items subject to constraints and receives a reward if and only if the weights of all chosen items are one. The weights of the items are binary, stochastic, and drawn independently of each other. The agent observes the index of the first chosen item whose weight is zero. This observation model arises in network routing, for instance, where the learning agent may only observe the first link in the routing path which is down, and blocks the path. We prop…
Carnegie Mellon University（卡内基梅隆大学）；Université de Montréal（蒙特利尔大学）；Adobe Research（Adobe研究院）
Most recommender systems recommend a list of items. The user examines the list, from the first item to the last, and often chooses the first attractive item and does not examine the rest. This type of user behavior can be modeled by the cascade model. In this work, we study cascading bandits, an online learning variant of the cascade model where the goal is to recommend $K$ most attractive items from a large set of $L$ candidate items. We propose two algorithms for solving this problem, which are based on the idea of linear generalization. The key idea in our solutions is that we learn a predi…
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…
We consider a new setting of online clustering of contextual cascading bandits, an online learning problem where the underlying cluster structure over users is unknown and needs to be learned from a random prefix feedback. More precisely, a learning agent recommends an ordered list of items to a user, who checks the list and stops at the first satisfactory item, if any. We propose an algorithm of CLUB-cascade for this setting and prove a $T$-step regret bound of order $\tilde{O}(\sqrt{T})$. Previous work corresponds to the degenerate case of only one cluster, and our general regret bound in th…
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…
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 …
Cascading Bandits Robust to Adversarial Corruptions
Hong Kong University of Science and Technology（香港科技大学）；East China Normal University（华东师范大学）；The Chinese University of Hong Kong（香港中文大学）；Shanghai Jiao Tong University（上海交通大学）
The Chinese University of Hong Kong（香港中文大学）；Shanghai Jiao Tong University（上海交通大学）；Tencent（腾讯）
The problem of multi-armed bandits (MAB) asks to make sequential decisions while balancing between exploitation and exploration, and have been successfully applied to a wide range of practical scenarios. Various algorithms have been designed to achieve a high reward in a long term. However, its short-term performance might be rather low, which is injurious in risk sensitive applications. Building on previous work of conservative bandits, we bring up a framework of contextual combinatorial conservative bandits. An algorithm is presented and a regret bound of $\tilde O(d^2+d\sqrt{T})$ is proven,…
Facebook AI Research（Facebook人工智能研究（Facebook AI Research））
In many fields such as digital marketing, healthcare, finance, and robotics, it is common to have a well-tested and reliable baseline policy running in production (e.g., a recommender system). Nonetheless, the baseline policy is often suboptimal. In this case, it is desirable to deploy online learning algorithms (e.g., a multi-armed bandit algorithm) that interact with the system to learn a better/optimal policy under the constraint that during the learning process the performance is almost never worse than the performance of the baseline itself. In this paper, we study the conservative learni…
We design and analyze CascadeBAI, an algorithm for finding the best set of $K$ items, also called an arm, within the framework of cascading bandits. An upper bound on the time complexity of CascadeBAI is derived by overcoming a crucial analytical challenge, namely, that of probabilistically estimating the amount of available feedback at each step. To do so, we define a new class of random variables (r.v.'s) which we term as left-sided sub-Gaussian r.v.'s; these are r.v.'s whose cumulant generating functions (CGFs) can be bounded by a quadratic only for non-positive arguments of the CGFs. This …
University of Science and Technology of China（中国科学技术大学）；The Pennsylvania State University（宾夕法尼亚州立大学）
In this paper, we propose a cost-aware cascading bandits model, a new variant of multi-armed ban- dits with cascading feedback, by considering the random cost of pulling arms. In each step, the learning agent chooses an ordered list of items and examines them sequentially, until certain stopping condition is satisfied. Our objective is then to max- imize the expected net reward in each step, i.e., the reward obtained in each step minus the total cost in- curred in examining the items, by deciding the or- dered list of items, as well as when to stop examina- tion. We study both the offline and …
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…
Safety is a desirable property that can immensely increase the applicability of learning algorithms in real-world decision-making problems. It is much easier for a company to deploy an algorithm that is safe, i.e., guaranteed to perform at least as well as a baseline. In this paper, we study the issue of safety in contextual linear bandits that have application in many different fields including personalized ad recommendation in online marketing. We formulate a notion of safety for this class of algorithms. We develop a safe contextual linear bandit algorithm, called conservative linear UCB (C…
In this paper, we study a family of conservative bandit problems (CBPs) with sample-path reward constraints, i.e., the learner's reward performance must be at least as well as a given baseline at any time. We propose a One-Size-Fits-All solution to CBPs and present its applications to three encompassed problems, i.e. conservative multi-armed bandits (CMAB), conservative linear bandits (CLB) and conservative contextual combinatorial bandits (CCCB). Different from previous works which consider high probability constraints on the expected reward, we focus on a sample-path constraint on the actual…
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 study a novel multi-armed bandit problem that models the challenge faced by a company wishing to explore new strategies to maximize revenue whilst simultaneously maintaining their revenue above a fixed baseline, uniformly over time. While previous work addressed the problem under the weaker requirement of maintaining the revenue constraint only at a given fixed time in the future, the algorithms previously proposed are unsuitable due to their design under the more stringent constraints. We consider both the stochastic and the adversarial settings, where we propose, natural, yet novel strate…
We analyze the regret of combinatorial Thompson sampling (CTS) for the combinatorial multi-armed bandit with probabilistically triggered arms under the semi-bandit feedback setting. We assume that the learner has access to an exact optimization oracle but does not know the expected base arm outcomes beforehand. When the expected reward function is Lipschitz continuous in the expected base arm outcomes, we derive $O(\sum_{i =1}^m \log T / (p_i Δ_i))$ regret bound for CTS, where $m$ denotes the number of base arms, $p_i$ denotes the minimum non-zero triggering probability of base arm $i$ and $Δ_…
Many physical systems have underlying safety considerations that require that the strategy deployed ensures the satisfaction of a set of constraints. Further, often we have only partial information on the state of the system. We study the problem of safe real-time decision making under uncertainty. In this paper, we formulate a conservative stochastic contextual bandit formulation for real-time decision making when an adversary chooses a distribution on the set of possible contexts and the learner is subject to certain safety/performance constraints. The learner observes only the context distr…
University of California, Santa Barbara（加利福尼亚大学圣巴巴拉分校）
We study stage-wise conservative linear stochastic bandits: an instance of bandit optimization, which accounts for (unknown) safety constraints that appear in applications such as online advertising and medical trials. At each stage, the learner must choose actions that not only maximize cumulative reward across the entire time horizon but further satisfy a linear baseline constraint that takes the form of a lower bound on the instantaneous reward. For this problem, we present two novel algorithms, stage-wise conservative linear Thompson Sampling (SCLTS) and stage-wise conservative linear UCB …
New York University（纽约大学）；Microsoft Research（微软研究院）；Inria（法国国家信息与自动化研究所）
Contextual bandit algorithms are essential for solving many real-world interactive machine learning problems. Despite multiple recent successes on statistically and computationally efficient methods, the practical behavior of these algorithms is still poorly understood. We leverage the availability of large numbers of supervised learning datasets to empirically evaluate contextual bandit algorithms, focusing on practical methods that learn by relying on optimization oracles from supervised learning. We find that a recent method (Foster et al., 2018) using optimism under uncertainty works the b…
Paderborn University（帕德博恩大学）；University of Munich (LMU)（慕尼黑大学）；Munich Center for Machine Learning（慕尼黑机器学习中心）
We consider the combinatorial bandits problem with semi-bandit feedback under finite sampling budget constraints, in which the learner can carry out its action only for a limited number of times specified by an overall budget. The action is to choose a set of arms, whereupon feedback for each arm in the chosen set is received. Unlike existing works, we study this problem in a non-stochastic setting with subset-dependent feedback, i.e., the semi-bandit feedback received could be generated by an oblivious adversary and also might depend on the chosen set of arms. In addition, we consider a gener…
Google Research（谷歌研究院）；DeepMind（深度思维）；University of Alberta（阿尔伯塔大学）
Most bandit policies are designed to either minimize regret in any problem instance, making very few assumptions about the underlying environment, or in a Bayesian sense, assuming a prior distribution over environment parameters. The former are often too conservative in practical settings, while the latter require assumptions that are hard to verify in practice. We study bandit problems that fall between these two extremes, where the learning agent has access to sampled bandit instances from an unknown prior distribution $\mathcal{P}$ and aims to achieve high reward on average over the bandit …
Adversarial Attacks on Combinatorial Multi-Armed Bandits
Oregon State University（俄勒冈州立大学）；University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Pennsylvania State University（宾夕法尼亚州立大学）；Princeton University（普林斯顿大学）
Google Research（谷歌研究院）；DeepMind（深度思维）；University of Alberta（阿尔伯塔大学）
Most bandit algorithm designs are purely theoretical. Therefore, they have strong regret guarantees, but also are often too conservative in practice. In this work, we pioneer the idea of algorithm design by minimizing the empirical Bayes regret, the average regret over problem instances sampled from a known distribution. We focus on a tractable instance of this problem, the confidence interval and posterior width tuning, and propose an efficient algorithm for solving it. The tuning algorithm is analyzed and evaluated in multi-armed, linear, and generalized linear bandits. We report several-fol…
Bandit algorithms have various application in safety-critical systems, where it is important to respect the system constraints that rely on the bandit's unknown parameters at every round. In this paper, we formulate a linear stochastic multi-armed bandit problem with safety constraints that depend (linearly) on an unknown parameter vector. As such, the learner is unable to identify all safe actions and must act conservatively in ensuring that her actions satisfy the safety constraint at all rounds (at least with high probability). For these bandits, we propose a new UCB-based algorithm called …
University of Pennsylvania（宾夕法尼亚大学）；University of Pittsburgh（匹兹堡大学）
Excessively changing policies in many real world scenarios is difficult, unethical, or expensive. After all, doctor guidelines, tax codes, and price lists can only be reprinted so often. We may thus want to only change a policy when it is probable that the change is beneficial. In cases that a policy is a threshold on contextual variables we can estimate treatment effects for populations lying at the threshold. This allows for a schedule of incremental policy updates that let us optimize a policy while making few detrimental changes. Using this idea, and the theory of linear contextual bandits…
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；Adobe Research（奥多比研究院）；Google Research（谷歌研究院）；Comcast Cable（康卡斯特电缆）
In many practical problems, a learning agent may want to learn the best action in hindsight without ever taking a bad action, which is significantly worse than the default production action. In general, this is impossible because the agent has to explore unknown actions, some of which can be bad, to learn better actions. However, when the actions are combinatorial, this may be possible if the unknown action can be evaluated by interleaving it with the production action. We formalize this concept as learning in stochastic combinatorial semi-bandits with exchangeable actions. We design efficient…
University of Washington（华盛顿大学）；Rubrik, Inc.（Rubrik公司）；Microsoft Research（微软研究院）
Recent advances have enabled "oracle" classifiers that can classify across many classes and input distributions with high accuracy without retraining. However, these classifiers are relatively heavyweight, so that applying them to classify video is costly. We show that day-to-day video exhibits highly skewed class distributions over the short term, and that these distributions can be classified by much simpler models. We formulate the problem of detecting the short-term skews online and exploiting models based on it as a new sequential decision making problem dubbed the Online Bandit Problem, …
Shanghai Jiao Tong University（上海交通大学）；University of Michigan（密歇根大学）；The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））
This paper studies \emph{differential privacy (DP)} and \emph{local differential privacy (LDP)} in cascading bandits. Under DP, we propose an algorithm which guarantees $ε$-indistinguishability and a regret of $\mathcal{O}((\frac{\log T}ε)^{1+ξ})$ for an arbitrarily small $ξ$. This is a significant improvement from the previous work of $\mathcal{O}(\frac{\log^3 T}ε)$ regret. Under ($ε$,$δ$)-LDP, we relax the $K^2$ dependence through the tradeoff between privacy budget $ε$ and error probability $δ$, and obtain a regret of $\mathcal{O}(\frac{K\log (1/δ) \log T}{ε^2})$, where $K$ is the size of t…
Contextual Combinatorial Bandits with Probabilistically Triggered Arms
The Chinese University of Hong Kong（香港中文大学）；University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校）；California Institute of Technology（加州理工学院）；Microsoft Research（微软研究院）
We study combinatorial multi-armed bandit with probabilistically triggered arms (CMAB-T) and semi-bandit feedback. We resolve a serious issue in the prior CMAB-T studies where the regret bounds contain a possibly exponentially large factor of $1/p^*$, where $p^*$ is the minimum positive probability that an arm is triggered by any action. We address this issue by introducing a triggering probability modulated (TPM) bounded smoothness condition into the general CMAB-T framework, and show that many applications such as influence maximization bandit and combinatorial cascading bandit satisfy this …
In this paper, we propose a new multi-armed bandit problem called the Gambler's Ruin Bandit Problem (GRBP). In the GRBP, the learner proceeds in a sequence of rounds, where each round is a Markov Decision Process (MDP) with two actions (arms): a continuation action that moves the learner randomly over the state space around the current state; and a terminal action that moves the learner directly into one of the two terminal states (goal and dead-end state). The current round ends when a terminal state is reached, and the learner incurs a positive reward only when the goal state is reached. The…
Real-Time Cascade Mitigation in Power Systems Using Influence Graph Improved by Reinforcement Learning
School of Mechanical and Electrical Engineering, Soochow University（苏州大学机械与电子工程学院）；Department of Computer Science, Iowa State University（计算机科学系，爱荷华州立大学）；Department of Mechanical Engineering, University of Akron（机械工程系，阿克伦大学）；Department of Computer and Information Technology, Purdue University（计算机与信息科技系，普渡大学）
Multi-Task Representation Learning for Conservative Linear Bandits
Department of Computer Science and Technology, Qingdao University（计算机科学与技术系，青岛大学）；Department of Electrical and Computer Engineering, Iowa State University（电气与计算机工程系，爱荷华州立大学）
