We study a multi-round welfare-maximising mechanism design problem in instances where agents do not know their values. On each round, a mechanism first assigns an allocation each to a set of agents and charges them a price; at the end of the round, the agents provide (stochastic) feedback to the mechanism for the allocation they received. This setting is motivated by applications in cloud markets and online advertising where an agent may know her value for an allocation only after experiencing it. Therefore, the mechanism needs to explore different allocations for each agent so that it can learn their values, while simultaneously attempting to find the socially optimal set of allocations. Our focus is on truthful and individually rational mechanisms which imitate the classical VCG mechanism in the long run. To that end, we first define three notions of regret for the welfare, the individual utilities of each agent and that of the mechanism. We show that these three terms are interdependent via an $Ω(T^{\frac{2}{3}})$ lower bound for the maximum of these three terms after $T$ rounds of allocations, and describe an algorithm which essentially achieves this rate. Our framework also provides flexibility to control the pricing scheme so as to trade-off between the agent and seller regrets. Next, we define asymptotic variants for the truthfulness and individual rationality requirements and provide asymptotic rates to quantify the degree to which both properties are satisfied by the proposed algorithm.
In causal bandit problems, the action set consists of interventions on variables of a causal graph. Several researchers have recently studied such bandit problems and pointed out their practical applications. However, all existing works rely on a restrictive and impractical assumption that the learner is given full knowledge of the causal graph structure upfront. In this paper, we develop novel causal bandit algorithms without knowing the causal graph. Our algorithms work well for causal trees, causal forests and a general class of causal graphs. The regret guarantees of our algorithms greatly…
National University of Singapore（新加坡国立大学）；Cornell University（康奈尔大学）
Consider a nonparametric contextual multi-arm bandit problem where each arm $a \in [K]$ is associated to a nonparametric reward function $f_a: [0,1] \to \mathbb{R}$ mapping from contexts to the expected reward. Suppose that there is a large set of arms, yet there is a simple but unknown structure amongst the arm reward functions, e.g. finite types or smooth with respect to an unknown metric space. We present a novel algorithm which learns data-driven similarities amongst the arms, in order to implement adaptive partitioning of the context-arm space for more efficient learning. We provide regre…
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）
Bandits with covariates, a.k.a. contextual bandits, address situations where optimal actions (or arms) at a given time $t$, depend on a context $x_t$, e.g., a new patient's medical history, a consumer's past purchases. While it is understood that the distribution of contexts might change over time, e.g., due to seasonalities, or deployment to new environments, the bulk of studies concern the most adversarial such changes, resulting in regret bounds that are often worst-case in nature. Covariate-shift on the other hand has been considered in classification as a middle-ground formalism that can …
Learning Dynamic Mechanisms in Unknown Environments: A Reinforcement Learning Approach
Yale University（耶鲁大学）；The Hong Kong University of Science and Technology（香港科技大学）；University of Chicago（芝加哥大学）；Purdue University（普渡大学）；Northwestern University（西北大学）；University of California, Berkeley（加州大学伯克利分校）
Princeton University（普林斯顿大学）；University of Southern California（南加利福尼亚大学）；Massachusetts Institute of Technology（麻省理工学院）
We consider the problem of learning in episodic finite-horizon Markov decision processes with an unknown transition function, bandit feedback, and adversarial losses. We propose an efficient algorithm that achieves $\mathcal{\tilde{O}}(L|X|\sqrt{|A|T})$ regret with high probability, where $L$ is the horizon, $|X|$ is the number of states, $|A|$ is the number of actions, and $T$ is the number of episodes. To the best of our knowledge, our algorithm is the first to ensure $\mathcal{\tilde{O}}(\sqrt{T})$ regret in this challenging setting; in fact it achieves the same regret bound as (Rosenberg &…
In this paper, we study a novel Stochastic Network Utility Maximization (NUM) problem where the utilities of agents are unknown. The utility of each agent depends on the amount of resource it receives from a network operator/controller. The operator desires to do a resource allocation that maximizes the expected total utility of the network. We consider threshold type utility functions where each agent gets non-zero utility if the amount of resource it receives is higher than a certain threshold. Otherwise, its utility is zero (hard real-time). We pose this NUM setup with unknown utilities as …
Contextual multi-armed bandits are classical models in reinforcement learning for sequential decision-making associated with individual information. A widely-used policy for bandits is Thompson Sampling, where samples from a data-driven probabilistic belief about unknown parameters are used to select the control actions. For this computationally fast algorithm, performance analyses are available under full context-observations. However, little is known for problems that contexts are not fully observed. We propose a Thompson Sampling algorithm for partially observable contextual multi-armed ban…
Wichita State University（威奇托州立大学）；Department of Electrical Engineering and Computer Science（电气工程与计算机科学系（不单独作为机构提取，此处主要机构为大学））
This paper addresses the problem of unknown delays in adversarial multi-armed bandit (MAB) with multiple play. Existing work on similar game setting focused on only the case where the learner selects an arm in each round. However, there are lots of applications in robotics where a learner needs to select more than one arm per round. It is therefore worthwhile to investigate the effect of delay when multiple arms are chosen. The multiple arms chosen per round in this setting are such that they experience the same amount of delay. There can be an aggregation of feedback losses from different com…
We consider the problem of model selection for two popular stochastic linear bandit settings, and propose algorithms that adapts to the unknown problem complexity. In the first setting, we consider the $K$ armed mixture bandits, where the mean reward of arm $i \in [K]$, is $μ_i+ \langle α_{i,t},θ^* \rangle $, with $α_{i,t} \in \mathbb{R}^d$ being the known context vector and $μ_i \in [-1,1]$ and $θ^*$ are unknown parameters. We define $\|θ^*\|$ as the problem complexity and consider a sequence of nested hypothesis classes, each positing a different upper bound on $\|θ^*\|$. Exploiting this, we…
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 …
Online Learning for Dynamic Vickrey-Clarke-Groves Mechanism in Unknown Environments
Department of Industrial \& Enterprise Systems Engineering；Coordinated Science Laboratory, University of Illinois Urbana-Champaign, Urbana, IL 61801, USA；Department of Industrial \& Enterprise Systems Engineering, Department of Electrical \& Computer Engineering
We consider Thompson sampling for linear bandit problems with finitely many independent arms, where rewards are sampled from normal distributions that are linearly dependent on unknown parameter vectors and with unknown variance. Specifically, with a Bayesian formulation we consider multivariate normal-gamma priors to represent environment uncertainty for all involved parameters. We show that our chosen sampling prior is a conjugate prior to the reward model and derive a Bayesian regret bound for Thompson sampling under the condition that the 5/2-moment of the variance distribution exist.
Princeton University（普林斯顿大学）；Google AI Princeton（谷歌人工智能普林斯顿）
We study the problem of controlling a linear dynamical system with adversarial perturbations where the only feedback available to the controller is the scalar loss, and the loss function itself is unknown. For this problem, with either a known or unknown system, we give an efficient sublinear regret algorithm. The main algorithmic difficulty is the dependence of the loss on past controls. To overcome this issue, we propose an efficient algorithm for the general setting of bandit convex optimization for loss functions with memory, which may be of independent interest.
The target of $\mathcal{X}$-armed bandit problem is to find the global maximum of an unknown stochastic function $f$, given a finite budget of $n$ evaluations. Recently, $\mathcal{X}$-armed bandits have been widely used in many situations. Many of these applications need to deal with large-scale data sets. To deal with these large-scale data sets, we study a distributed setting of $\mathcal{X}$-armed bandits, where $m$ players collaborate to find the maximum of the unknown function. We develop a novel anytime distributed $\mathcal{X}$-armed bandit algorithm. Compared with prior work on $\mathc…
The stochastic linear bandit problem proceeds in rounds where at each round the algorithm selects a vector from a decision set after which it receives a noisy linear loss parameterized by an unknown vector. The goal in such a problem is to minimize the (pseudo) regret which is the difference between the total expected loss of the algorithm and the total expected loss of the best fixed vector in hindsight. In this paper, we consider settings where the unknown parameter has structure, e.g., sparse, group sparse, low-rank, which can be captured by a norm, e.g., $L_1$, $L_{(1,2)}$, nuclear norm. W…
New York University（纽约大学）；University of Arizona（亚利桑那大学）；Microsoft Research（微软研究院）
We create a computationally tractable algorithm for contextual bandits with continuous actions having unknown structure. Our reduction-style algorithm composes with most supervised learning representations. We prove that it works in a general sense and verify the new functionality with large-scale experiments.
Technion - Israel Institute of Technology（以色列理工学院）；Tel Aviv University（特拉维夫大学）
Policy optimization methods are one of the most widely used classes of Reinforcement Learning (RL) algorithms. Yet, so far, such methods have been mostly analyzed from an optimization perspective, without addressing the problem of exploration, or by making strong assumptions on the interaction with the environment. In this paper we consider model-based RL in the tabular finite-horizon MDP setting with unknown transitions and bandit feedback. For this setting, we propose an optimistic trust region policy optimization (TRPO) algorithm for which we establish $\tilde O(\sqrt{S^2 A H^4 K})$ regret …
Consider the problem of a controller sampling sequentially from a finite number of $N \geq 2$ populations, specified by random variables $X^i_k$, $ i = 1,\ldots , N,$ and $k = 1, 2, \ldots$; where $X^i_k$ denotes the outcome from population $i$ the $k^{th}$ time it is sampled. It is assumed that for each fixed $i$, $\{ X^i_k \}_{k \geq 1}$ is a sequence of i.i.d. uniform random variables over some interval $[a_i, b_i]$, with the support (i.e., $a_i, b_i$) unknown to the controller. The objective is to have a policy $π$ for deciding, based on available data, from which of the $N$ populations to…
In the classic multi-armed bandits problem, the goal is to have a policy for dynamically operating arms that each yield stochastic rewards with unknown means. The key metric of interest is regret, defined as the gap between the expected total reward accumulated by an omniscient player that knows the reward means for each arm, and the expected total reward accumulated by the given policy. The policies presented in prior work have storage, computation and regret all growing linearly with the number of arms, which is not scalable when the number of arms is large. We consider in this work a broad …
In this paper, we study censored Semi-Bandits, a novel variant of the semi-bandits problem. The learner is assumed to have a fixed amount of resources, which it allocates to the arms at each time step. The loss observed from an arm is random and depends on the amount of resources allocated to it. More specifically, the loss equals zero if the allocation for the arm exceeds a constant (but unknown)threshold that can be dependent on the arm. Our goal is to learn a feasible allocation that minimizes the expected loss. The problem is challenging because the loss distribution and threshold value of…
Consider a decision-maker that can pick one out of $K$ actions to control an unknown system, for $T$ turns. The actions are interpreted as different configurations or policies. Holding the same action fixed, the system asymptotically converges to a unique equilibrium, as a function of this action. The dynamics of the system are unknown to the decision-maker, which can only observe a noisy reward at the end of every turn. The decision-maker wants to maximize its accumulated reward over the $T$ turns. Learning what equilibria are better results in higher rewards, but waiting for the system to co…
University of Cambridge（剑桥大学）；INRIA Lille - Nord Europe（法国国家信息与自动化研究所里尔-北欧中心）
We consider a stochastic bandit problem with infinitely many arms. In this setting, the learner has no chance of trying all the arms even once and has to dedicate its limited number of samples only to a certain number of arms. All previous algorithms for this setting were designed for minimizing the cumulative regret of the learner. In this paper, we propose an algorithm aiming at minimizing the simple regret. As in the cumulative regret setting of infinitely many armed bandits, the rate of the simple regret will depend on a parameter $β$ characterizing the distribution of the near-optimal arm…
University of California Berkeley（加州大学伯克利分校）；University of Massachusetts（马萨诸塞大学）；DeepMind（深度思维）；Adobe（奥多比）；Google Research（谷歌研究院）；University of Alberta（阿尔伯塔大学）
We study bandit model selection in stochastic environments. Our approach relies on a meta-algorithm that selects between candidate base algorithms. We develop a meta-algorithm-base algorithm abstraction that can work with general classes of base algorithms and different type of adversarial meta-algorithms. Our methods rely on a novel and generic smoothing transformation for bandit algorithms that permits us to obtain optimal $O(\sqrt{T})$ model selection guarantees for stochastic contextual bandit problems as long as the optimal base algorithm satisfies a high probability regret guarantee. We …
In a linear stochastic bandit model, each arm is a vector in an Euclidean space and the observed return at each time step is an unknown linear function of the chosen arm at that time step. In this paper, we investigate the problem of learning the best arm in a linear stochastic bandit model, where each arm's expected reward is an unknown linear function of the projection of the arm onto a subspace. We call this the projection reward. Unlike the classical linear bandit problem in which the observed return corresponds to the reward, the projection reward at each time step is unobservable. Such a…
Italian Institute of Technology（意大利理工学院）；Google Research（谷歌研究院）；University College London（伦敦大学学院）
Motivated by a natural problem in online model selection with bandit information, we introduce and analyze a best arm identification problem in the rested bandit setting, wherein arm expected losses decrease with the number of times the arm has been played. The shape of the expected loss functions is similar across arms, and is assumed to be available up to unknown parameters that have to be learned on the fly. We define a novel notion of regret for this problem, where we compare to the policy that always plays the arm having the smallest expected loss at the end of the game. We analyze an arm…
In algorithm optimization in reinforcement learning, how to deal with the exploration-exploitation dilemma is particularly important. Multi-armed bandit problem can optimize the proposed solutions by changing the reward distribution to realize the dynamic balance between exploration and exploitation. Thompson Sampling is a common method for solving multi-armed bandit problem and has been used to explore data that conform to various laws. In this paper, we consider the Thompson Sampling approach for multi-armed bandit problem, in which rewards conform to unknown asymmetric $α$-stable distributi…
Massachusetts Institute of Technology（麻省理工学院）；Google Research（谷歌研究院）；Courant Institute of Mathematical Sciences（柯朗数学科学研究所）
A major research direction in contextual bandits is to develop algorithms that are computationally efficient, yet support flexible, general-purpose function approximation. Algorithms based on modeling rewards have shown strong empirical performance, but typically require a well-specified model, and can fail when this assumption does not hold. Can we design algorithms that are efficient and flexible, yet degrade gracefully in the face of model misspecification? We introduce a new family of oracle-efficient algorithms for $\varepsilon$-misspecified contextual bandits that adapt to unknown model …
University of California at Davis（加州大学戴维斯分校）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
We study contextual bandits with budget and time constraints, referred to as constrained contextual bandits.The time and budget constraints significantly complicate the exploration and exploitation tradeoff because they introduce complex coupling among contexts over time.Such coupling effects make it difficult to obtain oracle solutions that assume known statistics of bandits. To gain insight, we first study unit-cost systems with known context distribution. When the expected rewards are known, we develop an approximation of the oracle, referred to Adaptive-Linear-Programming (ALP), which achi…
University of California San Diego（加州大学圣地亚哥分校）；University of Arizona（亚利桑那大学）
In many real-world applications, multiple agents seek to learn how to perform highly related yet slightly different tasks in an online bandit learning protocol. We formulate this problem as the $ε$-multi-player multi-armed bandit problem, in which a set of players concurrently interact with a set of arms, and for each arm, the reward distributions for all players are similar but not necessarily identical. We develop an upper confidence bound-based algorithm, RobustAgg$(ε)$, that adaptively aggregates rewards collected by different players. In the setting where an upper bound on the pairwise si…
We study a non-parametric multi-armed bandit problem with stochastic covariates, where a key complexity driver is the smoothness of payoff functions with respect to covariates. Previous studies have focused on deriving minimax-optimal algorithms in cases where it is a priori known how smooth the payoff functions are. In practice, however, the smoothness of payoff functions is typically not known in advance, and misspecification of smoothness may severely deteriorate the performance of existing methods. In this work, we consider a framework where the smoothness of payoff functions is not known,…
Indian Institute of Technology Bombay（孟买印度理工学院）；Indian Institute of Technology Madras（马德拉斯印度理工学院）；LinkedIn India（领英印度）
We consider the problem of sequentially allocating resources in a censored semi-bandits setup, where the learner allocates resources at each step to the arms and observes loss. The loss depends on two hidden parameters, one specific to the arm but independent of the resource allocation, and the other depends on the allocated resource. More specifically, the loss equals zero for an arm if the resource allocated to it exceeds a constant (but unknown) arm dependent threshold. The goal is to learn a resource allocation that minimizes the expected loss. The problem is challenging because the loss d…
Technicolor Labs（特艺集团实验室）；Yahoo Labs（雅虎实验室）；INRIA Lille - Nord Europe（法国国立信息学及其应用研究院里尔-北欧研究中心）
Many important optimization problems, such as the minimum spanning tree and minimum-cost flow, can be solved optimally by a greedy method. In this work, we study a learning variant of these problems, where the model of the problem is unknown and has to be learned by interacting repeatedly with the environment in the bandit setting. We formalize our learning problem quite generally, as learning how to maximize an unknown modular function on a known polymatroid. We propose a computationally efficient algorithm for solving our problem and bound its expected cumulative regret. Our gap-dependent up…
We consider the problem of finitely parameterized multi-armed bandits where the model of the underlying stochastic environment can be characterized based on a common unknown parameter. The true parameter is unknown to the learning agent. However, the set of possible parameters, which is finite, is known a priori. We propose an algorithm that is simple and easy to implement, which we call Finitely Parameterized Upper Confidence Bound (FP-UCB) algorithm, which uses the information about the underlying parameter set for faster learning. In particular, we show that the FP-UCB algorithm achieves a …
We consider the restless multi-armed bandit (RMAB) problem with unknown dynamics in which a player chooses M out of N arms to play at each time. The reward state of each arm transits according to an unknown Markovian rule when it is played and evolves according to an arbitrary unknown random process when it is passive. The performance of an arm selection policy is measured by regret, defined as the reward loss with respect to the case where the player knows which M arms are the most rewarding and always plays the M best arms. We construct a policy with an interleaving exploration and exploitat…
We study the problem of estimating the value of a known smooth function $f$ at an unknown point $\boldsymbolμ \in \mathbb{R}^n$, where each component $μ_i$ can be sampled via a noisy oracle. Sampling more frequently components of $\boldsymbolμ$ corresponding to directions of the function with larger directional derivatives is more sample-efficient. However, as $\boldsymbolμ$ is unknown, the optimal sampling frequencies are also unknown. We design an instance-adaptive algorithm that learns to sample according to the importance of each coordinate, and with probability at least $1-δ$ returns an $…
Tel Aviv University（特拉维夫大学）；Google Research（谷歌研究院）
We study agents communicating over an underlying network by exchanging messages, in order to optimize their individual regret in a common nonstochastic multi-armed bandit problem. We derive regret minimization algorithms that guarantee for each agent $v$ an individual expected regret of $\widetilde{O}\left(\sqrt{\left(1+\frac{K}{\left|\mathcal{N}\left(v\right)\right|}\right)T}\right)$, where $T$ is the number of time steps, $K$ is the number of actions and $\mathcal{N}\left(v\right)$ is the set of neighbors of agent $v$ in the communication graph. We present algorithms both for the case that t…
We consider a dynamic pricing problem under unknown demand models. In this problem a seller offers prices to a stream of customers and observes either success or failure in each sale attempt. The underlying demand model is unknown to the seller and can take one of N possible forms. In this paper, we show that this problem can be formulated as a multi-armed bandit with dependent arms. We propose a dynamic pricing policy based on the likelihood ratio test. We show that the proposed policy achieves complete learning, i.e., it offers a bounded regret where regret is defined as the revenue loss wit…
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…
Microsoft Research（微软研究院）；University of Arizona（亚利桑那大学）
We study contextual bandit learning with an abstract policy class and continuous action space. We obtain two qualitatively different regret bounds: one competes with a smoothed version of the policy class under no continuity assumptions, while the other requires standard Lipschitz assumptions. Both bounds exhibit data-dependent "zooming" behavior and, with no tuning, yield improved guarantees for benign problems. We also study adapting to unknown smoothness parameters, establishing a price-of-adaptivity and deriving optimal adaptive algorithms that require no additional information.
A classic setting of the stochastic K-armed bandit problem is considered in this note. In this problem it has been known that KL-UCB policy achieves the asymptotically optimal regret bound and KL-UCB+ policy empirically performs better than the KL-UCB policy although the regret bound for the original form of the KL-UCB+ policy has been unknown. This note demonstrates that a simple proof of the asymptotic optimality of the KL-UCB+ policy can be given by the same technique as those used for analyses of other known policies.
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 …
