We consider a single-hop wireless network with sources transmitting time-sensitive information to the destination over multiple unreliable channels. Packets from each source are generated according to a stochastic process with known statistics and the state of each wireless channel (ON/OFF) varies according to a stochastic process with unknown statistics. The reliability of the wireless channels is to be learned through observation. At every time slot, the learning algorithm selects a single pair (source, channel) and the selected source attempts to transmit its packet via the selected channel. The probability of a successful transmission to the destination depends on the reliability of the selected channel. The goal of the learning algorithm is to minimize the Age-of-Information (AoI) in the network over $T$ time slots. To analyze the performance of the learning algorithm, we introduce the notion of AoI regret, which is the difference between the expected cumulative AoI of the learning algorithm under consideration and the expected cumulative AoI of a genie algorithm that knows the reliability of the channels a priori. The AoI regret captures the penalty incurred by having to learn the statistics of the channels over the $T$ time slots. The results are two-fold: first, we consider learning algorithms that employ well-known solutions to the stochastic multi-armed bandit problem (such as $ε$-Greedy, Upper Confidence Bound, and Thompson Sampling) and show that their AoI regret scales as $Θ(\log T)$; second, we develop a novel learning algorithm and show that it has $O(1)$ regret. To the best of our knowledge, this is the first learning algorithm with bounded AoI regret.
We prove an instance independent (poly) logarithmic regret for stochastic contextual bandits with linear payoff. Previously, in \cite{chu2011contextual}, a lower bound of $\mathcal{O}(\sqrt{T})$ is shown for the contextual linear bandit problem with arbitrary (adversarily chosen) contexts. In this paper, we show that stochastic contexts indeed help to reduce the regret from $\sqrt{T}$ to $\polylog(T)$. We propose Low Regret Stochastic Contextual Bandits (\texttt{LR-SCB}), which takes advantage of the stochastic contexts and performs parameter estimation (in $\ell_2$ norm) and regret minimizati…
Montanuniversitaet Leoben（莱奥本矿业大学）；University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
We present an algorithm that achieves almost optimal pseudo-regret bounds against adversarial and stochastic bandits. Against adversarial bandits the pseudo-regret is $O(K\sqrt{n \log n})$ and against stochastic bandits the pseudo-regret is $O(\sum_i (\log n)/Δ_i)$. We also show that no algorithm with $O(\log n)$ pseudo-regret against stochastic bandits can achieve $\tilde{O}(\sqrt{n})$ expected regret against adaptive adversarial bandits. This complements previous results of Bubeck and Slivkins (2012) that show $\tilde{O}(\sqrt{n})$ expected adversarial regret with $O((\log n)^2)$ stochastic …
An adversarial bandit problem with memory constraints is studied where only the statistics of a subset of arms can be stored. A hierarchical learning policy that requires only a sublinear order of memory space in terms of the number of arms is developed. Its sublinear regret orders with respect to the time horizon are established for both weak regret and shifting regret. This work appears to be the first on memory-constrained bandit problems under the adversarial setting.
Toyota Technological Institute at Chicago（芝加哥丰田理工学院）；Microsoft Research（微软研究院）
Online learning algorithms are designed to learn even when their input is generated by an adversary. The widely-accepted formal definition of an online algorithm's ability to learn is the game-theoretic notion of regret. We argue that the standard definition of regret becomes inadequate if the adversary is allowed to adapt to the online algorithm's actions. We define the alternative notion of policy regret, which attempts to provide a more meaningful way to measure an online algorithm's performance against adaptive adversaries. Focusing on the online bandit setting, we show that no bandit algo…
In this paper, we study the multi-armed bandit problem in the batched setting where the employed policy must split data into a small number of batches. While the minimax regret for the two-armed stochastic bandits has been completely characterized in \cite{perchet2016batched}, the effect of the number of arms on the regret for the multi-armed case is still open. Moreover, the question whether adaptively chosen batch sizes will help to reduce the regret also remains underexplored. In this paper, we propose the BaSE (batched successive elimination) policy to achieve the rate-optimal regrets (wit…
We study regret minimization in a stochastic multi-armed bandit setting and establish a fundamental trade-off between the regret suffered under an algorithm, and its statistical robustness. Considering broad classes of underlying arms' distributions, we show that bandit learning algorithms with logarithmic regret are always inconsistent and that consistent learning algorithms always suffer a super-logarithmic regret. This result highlights the inevitable statistical fragility of all `logarithmic regret' bandit algorithms available in the literature---for instance, if a UCB algorithm designed f…
PSL - Université Paris Dauphine（巴黎多芬大学（PSL研究大学））；CNRS（法国国家科学研究中心）；Huawei Noah’s Ark Lab（华为诺亚方舟实验室）
We propose the first regret-based approach to the Graphical Bilinear Bandits problem, where $n$ agents in a graph play a stochastic bilinear bandit game with each of their neighbors. This setting reveals a combinatorial NP-hard problem that prevents the use of any existing regret-based algorithm in the (bi-)linear bandit literature. In this paper, we fill this gap and present the first regret-based algorithm for graphical bilinear bandits using the principle of optimism in the face of uncertainty. Theoretical analysis of this new method yields an upper bound of $\tilde{O}(\sqrt{T})$ on the $α$…
We revisit the proof by Qin et al. (2014) of bounded regret of the C$^2$UCB contextual combinatorial bandit. We demonstrate an error in the proof of volumetric expansion of the moment matrix, used in upper bounding a function of context vector norms. We prove a relaxed inequality that yields the originally-stated regret bound.
We study the adversarial online learning problem and create a completely online algorithmic framework that has data dependent regret guarantees in both full expert feedback and bandit feedback settings. We study the expected performance of our algorithm against general comparators, which makes it applicable for a wide variety of problem scenarios. Our algorithm works from a universal prediction perspective and the performance measure used is the expected regret against arbitrary comparator sequences, which is the difference between our losses and a competing loss sequence. The competition clas…
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；Microsoft Research（微软研究院）
Designing efficient general-purpose contextual bandit algorithms that work with large -- or even continuous -- action spaces would facilitate application to important scenarios such as information retrieval, recommendation systems, and continuous control. While obtaining standard regret guarantees can be hopeless, alternative regret notions have been proposed to tackle the large action setting. We propose a smooth regret notion for contextual bandits, which dominates previously proposed alternatives. We design a statistically and computationally efficient algorithm -- for the proposed smooth r…
We consider a variant of the contextual bandit problem. In standard contextual bandits, when a user arrives we get the user's complete feature vector and then assign a treatment (arm) to that user. In a number of applications (like healthcare), collecting features from users can be costly. To address this issue, we propose algorithms that avoid needless feature collection while maintaining strong regret guarantees.
We study the problem of regret minimization for distributed bandits learning, in which $M$ agents work collaboratively to minimize their total regret under the coordination of a central server. Our goal is to design communication protocols with near-optimal regret and little communication cost, which is measured by the total amount of transmitted data. For distributed multi-armed bandits, we propose a protocol with near-optimal regret and only $O(M\log(MK))$ communication cost, where $K$ is the number of arms. The communication cost is independent of the time horizon $T$, has only logarithmic …
Kernel-based bandit is an extensively studied black-box optimization problem, in which the objective function is assumed to live in a known reproducing kernel Hilbert space. While nearly optimal regret bounds (up to logarithmic factors) are established in the noisy setting, surprisingly, less is known about the noise-free setting (when the exact values of the underlying function is accessible without observation noise). We discuss several upper bounds on regret; none of which seem order optimal, and provide a conjecture on the order optimal regret bound.
The University of Tokyo（东京大学）；Kyoto University（京都大学）
We study the $K$-armed dueling bandit problem, a variation of the standard stochastic bandit problem where the feedback is limited to relative comparisons of a pair of arms. We introduce a tight asymptotic regret lower bound that is based on the information divergence. An algorithm that is inspired by the Deterministic Minimum Empirical Divergence algorithm (Honda and Takemura, 2010) is proposed, and its regret is analyzed. The proposed algorithm is found to be the first one with a regret upper bound that matches the lower bound. Experimental comparisons of dueling bandit algorithms show that …
We study the K-armed dueling bandit problem, a variation of the standard stochastic bandit problem where the feedback is limited to relative comparisons of a pair of arms. The hardness of recommending Copeland winners, the arms that beat the greatest number of other arms, is characterized by deriving an asymptotic regret bound. We propose Copeland Winners Relative Minimum Empirical Divergence (CW-RMED) and derive an asymptotically optimal regret bound for it. However, it is not known whether the algorithm can be efficiently computed or not. To address this issue, we devise an efficient version…
We study a nonparametric contextual bandit problem where the expected reward functions belong to a Hölder class with smoothness parameter $β$. We show how this interpolates between two extremes that were previously studied in isolation: non-differentiable bandits ($β\leq1$), where rate-optimal regret is achieved by running separate non-contextual bandits in different context regions, and parametric-response bandits (satisfying $β=\infty$), where rate-optimal regret can be achieved with minimal or no exploration due to infinite extrapolatability. We develop a novel algorithm that carefully adju…
Department of EECS, University of Michigan, Ann Arbor, MI, USA（密歇根大学电气工程与计算机科学系）；Department of Aerospace Engineering, University of Michigan, Ann Arbor, MI, USA（密歇根大学航空航天工程系）；Climate and Space Engineering, University of Michigan, Ann Arbor, MI, USA（密歇根大学气候与空间工程系）
There are two variants of the classical multi-armed bandit (MAB) problem that have received considerable attention from machine learning researchers in recent years: contextual bandits and simple regret minimization. Contextual bandits are a sub-class of MABs where, at every time step, the learner has access to side information that is predictive of the best arm. Simple regret minimization assumes that the learner only incurs regret after a pure exploration phase. In this work, we study simple regret minimization for contextual bandits. Motivated by applications where the learner has separate …
We consider a linear stochastic bandit problem involving $M$ agents that can collaborate via a central server to minimize regret. A fraction $α$ of these agents are adversarial and can act arbitrarily, leading to the following tension: while collaboration can potentially reduce regret, it can also disrupt the process of learning due to adversaries. In this work, we provide a fundamental understanding of this tension by designing new algorithms that balance the exploration-exploitation trade-off via carefully constructed robust confidence intervals. We also complement our algorithms with tight …
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 …
As noted in the works of \cite{lattimore2020bandit}, it has been mentioned that it is an open problem to characterize the minimax regret of linear bandits in a wide variety of action spaces. In this article we present an optimal regret lower bound for a wide class of convex action spaces.
Institute of Computing Technology, Chinese Academy of Sciences（中国科学院计算技术研究所）；University of Chinese Academy of Sciences（中国科学院大学）；Peking University（北京大学）
Multi-arm bandit (MAB) and stochastic linear bandit (SLB) are important models in reinforcement learning, and it is well-known that classical algorithms for bandits with time horizon $T$ suffer $Ω(\sqrt{T})$ regret. In this paper, we study MAB and SLB with quantum reward oracles and propose quantum algorithms for both models with $O(\mbox{poly}(\log T))$ regrets, exponentially improving the dependence in terms of $T$. To the best of our knowledge, this is the first provable quantum speedup for regrets of bandit problems and in general exploitation in reinforcement learning. Compared to previou…
Princeton University（普林斯顿大学）；Université Paris Diderot（巴黎狄德罗大学）
We study the stochastic multi-armed bandit problem when one knows the value $μ^{(\star)}$ of an optimal arm, as a well as a positive lower bound on the smallest positive gap $Δ$. We propose a new randomized policy that attains a regret {\em uniformly bounded over time} in this setting. We also prove several lower bounds, which show in particular that bounded regret is not possible if one only knows $Δ$, and bounded regret of order $1/Δ$ is not possible if one only knows $μ^{(\star)}$
Efficient Swap Regret Minimization in Combinatorial Bandits
National Technical University of Athens（雅典国立技术大学）；National and Kapodistrian University of Athens（雅典国立卡波季斯特里安大学）；Archimedes, Athena Research Center（雅典娜研究中心阿基米德部）；Univ. Grenoble Alpes（格勒诺布尔阿尔卑斯大学）；CNRS（法国国家科学研究中心）；Inria（法国国家信息与自动化研究所）；Grenoble INP（格勒诺布尔国立高等物理、数学与信息学院）；University of California, Irvine（加州大学尔湾分校）
DeepMind（深度思维）；UC Berkeley（加州大学伯克利分校）；University of Massachusetts（马萨诸塞大学）
We consider model selection in stochastic bandit and reinforcement learning problems. Given a set of base learning algorithms, an effective model selection strategy adapts to the best learning algorithm in an online fashion. We show that by estimating the regret of each algorithm and playing the algorithms such that all empirical regrets are ensured to be of the same order, the overall regret balancing strategy achieves a regret that is close to the regret of the optimal base algorithm. Our strategy requires an upper bound on the optimal base regret as input, and the performance of the strateg…
Univ. Grenoble Alpes（格勒诺布尔阿尔卑斯大学）；Inria（法国国家信息与自动化研究所）；CNRS（法国国家科学研究中心）；Grenoble INP（格勒诺布尔国立理工学院）；Toyota Technological Institute at Chicago（芝加哥丰田技术研究所）；IBM Research（IBM研究院）；University of Pennsylvania（宾夕法尼亚大学）
We address the problem of \emph{`Internal Regret'} in \emph{Sleeping Bandits} in the fully adversarial setup, as well as draw connections between different existing notions of sleeping regrets in the multiarmed bandits (MAB) literature and consequently analyze the implications: Our first contribution is to propose the new notion of \emph{Internal Regret} for sleeping MAB. We then proposed an algorithm that yields sublinear regret in that measure, even for a completely adversarial sequence of losses and availabilities. We further show that a low sleeping internal regret always implies a low ext…
I present the first algorithm for stochastic finite-armed bandits that simultaneously enjoys order-optimal problem-dependent regret and worst-case regret. Besides the theoretical results, the new algorithm is simple, efficient and empirically superb. The approach is based on UCB, but with a carefully chosen confidence parameter that optimally balances the risk of failing confidence intervals against the cost of excessive optimism.
University of Southern California（南加利福尼亚大学）；Nanjing University（南京大学）
We consider the problem of combining and learning over a set of adversarial bandit algorithms with the goal of adaptively tracking the best one on the fly. The CORRAL algorithm of Agarwal et al. (2017) and its variants (Foster et al., 2020a) achieve this goal with a regret overhead of order $\widetilde{O}(\sqrt{MT})$ where $M$ is the number of base algorithms and $T$ is the time horizon. The polynomial dependence on $M$, however, prevents one from applying these algorithms to many applications where $M$ is poly$(T)$ or even larger. Motivated by this issue, we propose a new recipe to corral a l…
Rice University（莱斯大学）；University of Arizona（亚利桑那大学）；University of Washington（华盛顿大学）
In this work we consider the problem of regret minimization for logistic bandits. The main challenge of logistic bandits is reducing the dependence on a potentially large problem dependent constant $κ$ that can at worst scale exponentially with the norm of the unknown parameter $θ_{\ast}$. Abeille et al. (2021) have applied self-concordance of the logistic function to remove this worst-case dependence providing regret guarantees like $O(d\log^2(κ)\sqrt{\dotμT}\log(|\mathcal{X}|))$ where $d$ is the dimensionality, $T$ is the time horizon, and $\dotμ$ is the variance of the best-arm. This work i…
On the Benefits of Free Exploration for Regret Minimization in Multi-Armed Bandits
Department of Mathematics, National University of Singapore（新加坡国立大学数学系）；Department of Mathematics, Department of Electrical and Computer Engineering, National University of Singapore（新加坡国立大学数学系、电子与计算机工程系）
We prove that the information-theoretic upper bound on the minimax regret for zeroth-order adversarial bandit convex optimisation is at most $O(d^{2.5} \sqrt{n} \log(n))$, where $d$ is the dimension and $n$ is the number of interactions. This improves on $O(d^{9.5} \sqrt{n} \log(n)^{7.5}$ by Bubeck et al. (2017). The proof is based on identifying an improved exploratory distribution for convex functions.
In this paper we propose a novel experimental design-based algorithm to minimize regret in online stochastic linear and combinatorial bandits. While existing literature tends to focus on optimism-based algorithms--which have been shown to be suboptimal in many cases--our approach carefully plans which action to take by balancing the tradeoff between information gain and reward, overcoming the failures of optimism. In addition, we leverage tools from the theory of suprema of empirical processes to obtain regret guarantees that scale with the Gaussian width of the action set, avoiding wasteful u…
Policy regret is a well established notion of measuring the performance of an online learning algorithm against an adaptive adversary. We study restrictions on the adversary that enable efficient minimization of the \emph{complete policy regret}, which is the strongest possible version of policy regret. We identify a gap in the current theoretical understanding of what sorts of restrictions permit tractability in this challenging setting. To resolve this gap, we consider a generalization of the stochastic multi armed bandit, which we call the \emph{tallying bandit}. This is an online learning …
Indiana University at Bloomington（印第安纳大学伯明顿分校）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
We consider the thresholding bandit problem, whose goal is to find arms of mean rewards above a given threshold $θ$, with a fixed budget of $T$ trials. We introduce LSA, a new, simple and anytime algorithm that aims to minimize the aggregate regret (or the expected number of mis-classified arms). We prove that our algorithm is instance-wise asymptotically optimal. We also provide comprehensive empirical results to demonstrate the algorithm's superior performance over existing algorithms under a variety of different scenarios.
Restless bandit problems are instances of non-stationary multi-armed bandits. These problems have been studied well from the optimization perspective, where the goal is to efficiently find a near-optimal policy when system parameters are known. However, very few papers adopt a learning perspective, where the parameters are unknown. In this paper, we analyze the performance of Thompson sampling in episodic restless bandits with unknown parameters. We consider a general policy map to define our competitor and prove an $\tilde{\mathcal{O}}(\sqrt{T})$ Bayesian regret bound. Our competitor is flexi…
We consider online content recommendation with implicit feedback through pairwise comparisons, formalized as the so-called dueling bandit problem. We study the dueling bandit problem in the Condorcet winner setting, and consider two notions of regret: the more well-studied strong regret, which is 0 only when both arms pulled are the Condorcet winner; and the less well-studied weak regret, which is 0 if either arm pulled is the Condorcet winner. We propose a new algorithm for this problem, Winner Stays (WS), with variations for each kind of regret: WS for weak regret (WS-W) has expected cumulat…
University of Alberta（阿尔伯塔大学）；INRIA（法国国家信息与自动化研究所）
We study a new type of K-armed bandit problem where the expected return of one arm may depend on the returns of other arms. We present a new algorithm for this general class of problems and show that under certain circumstances it is possible to achieve finite expected cumulative regret. We also give problem-dependent lower bounds on the cumulative regret showing that at least in special cases the new algorithm is nearly optimal.
Yahoo Research（雅虎研究院）；Stony Brook University（石溪大学）；University of California, San Diego（加利福尼亚大学圣迭戈分校）
We present an efficient second-order algorithm with $\tilde{O}(\frac{1}η\sqrt{T})$ regret for the bandit online multiclass problem. The regret bound holds simultaneously with respect to a family of loss functions parameterized by $η$, for a range of $η$ restricted by the norm of the competitor. The family of loss functions ranges from hinge loss ($η=0$) to squared hinge loss ($η=1$). This provides a solution to the open problem of (J. Abernethy and A. Rakhlin. An efficient bandit algorithm for $\sqrt{T}$-regret in online multiclass prediction? In COLT, 2009). We test our algorithm experimental…
University of California, Berkeley（加州大学伯克利分校）；Google Research（谷歌研究院）
We propose a simple model selection approach for algorithms in stochastic bandit and reinforcement learning problems. As opposed to prior work that (implicitly) assumes knowledge of the optimal regret, we only require that each base algorithm comes with a candidate regret bound that may or may not hold during all rounds. In each round, our approach plays a base algorithm to keep the candidate regret bounds of all remaining base algorithms balanced, and eliminates algorithms that violate their candidate bound. We prove that the total regret of this approach is bounded by the best valid candidat…
Microsoft Research（微软研究院）；Stanford University（斯坦福大学）；University of Southern California（南加州大学）
We study adaptive regret bounds in terms of the variation of the losses (the so-called path-length bounds) for both multi-armed bandit and more generally linear bandit. We first show that the seemingly suboptimal path-length bound of (Wei and Luo, 2018) is in fact not improvable for adaptive adversary. Despite this negative result, we then develop two new algorithms, one that strictly improves over (Wei and Luo, 2018) with a smaller path-length measure, and the other which improves over (Wei and Luo, 2018) for oblivious adversary when the path-length is large. Our algorithms are based on the w…
CNRS（法国国家科学研究中心）；Sorbonne Université（索邦大学）；Université Paris Diderot（巴黎狄德罗大学）；ENS Cachan（卡尚高等师范学校）；Université Paris-Saclay（巴黎-萨克雷大学）；Criteo AI Lab（Criteo人工智能实验室）
We consider the classical stochastic multi-armed bandit but where, from time to time and roughly with frequency $ε$, an extra observation is gathered by the agent for free. We prove that, no matter how small $ε$ is the agent can ensure a regret uniformly bounded in time. More precisely, we construct an algorithm with a regret smaller than $\sum_i \frac{\log(1/ε)}{Δ_i}$, up to multiplicative constant and loglog terms. We also prove a matching lower-bound, stating that no reasonable algorithm can outperform this quantity.
I analyse the frequentist regret of the famous Gittins index strategy for multi-armed bandits with Gaussian noise and a finite horizon. Remarkably it turns out that this approach leads to finite-time regret guarantees comparable to those available for the popular UCB algorithm. Along the way I derive finite-time bounds on the Gittins index that are asymptotically exact and may be of independent interest. I also discuss some computational issues and present experimental results suggesting that a particular version of the Gittins index strategy is a modest improvement on existing algorithms with…
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…
CREST（法国经济与统计研究中心）；ENSAE Paris（巴黎国立统计与经济管理学院）；London School of Economics（伦敦政治经济学院）；Criteo AI Lab（Criteo人工智能实验室）
Finding an optimal matching in a weighted graph is a standard combinatorial problem. We consider its semi-bandit version where either a pair or a full matching is sampled sequentially. We prove that it is possible to leverage a rank-1 assumption on the adjacency matrix to reduce the sample complexity and the regret of off-the-shelf algorithms up to reaching a linear dependency in the number of vertices (up to poly log terms).
Northwestern University（西北大学）；University of Florida（佛罗里达大学）；New York University（纽约大学）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Linear contextual bandit is an important class of sequential decision making problems with a wide range of applications to recommender systems, online advertising, healthcare, and many other machine learning related tasks. While there is a lot of prior research, tight regret bounds of linear contextual bandit with infinite action sets remain open. In this paper, we address this open problem by considering the linear contextual bandit with (changing) infinite action sets. We prove a regret upper bound on the order of $O(\sqrt{d^2T\log T})\times \text{poly}(\log\log T)$ where $d$ is the domain d…
We consider a policy gradient algorithm applied to a finite-arm bandit problem with Bernoulli rewards. We allow learning rates to depend on the current state of the algorithm, rather than use a deterministic time-decreasing learning rate. The state of the algorithm forms a Markov chain on the probability simplex. We apply Foster-Lyapunov techniques to analyse the stability of this Markov chain. We prove that if learning rates are well chosen then the policy gradient algorithm is a transient Markov chain and the state of the chain converges on the optimal arm with logarithmic or poly-logarithmi…
Northwestern University（西北大学）；University of Florida（佛罗里达大学）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
We study the linear contextual bandit problem with finite action sets. When the problem dimension is $d$, the time horizon is $T$, and there are $n \leq 2^{d/2}$ candidate actions per time period, we (1) show that the minimax expected regret is $Ω(\sqrt{dT (\log T) (\log n)})$ for every algorithm, and (2) introduce a Variable-Confidence-Level (VCL) SupLinUCB algorithm whose regret matches the lower bound up to iterated logarithmic factors. Our algorithmic result saves two $\sqrt{\log T}$ factors from previous analysis, and our information-theoretical lower bound also improves previous results …
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…
Contextual bandits are a rich model for sequential decision making given side information, with important applications, e.g., in recommender systems. We propose novel algorithms for contextual bandits harnessing neural networks to approximate the unknown reward function. We resolve the open problem of proving sublinear regret bounds in this setting for general context sequences, considering both fully-connected and convolutional networks. To this end, we first analyze NTK-UCB, a kernelized bandit optimization algorithm employing the Neural Tangent Kernel (NTK), and bound its regret in terms of…
Given a multi-armed bandit problem it may be desirable to achieve a smaller-than-usual worst-case regret for some special actions. I show that the price for such unbalanced worst-case regret guarantees is rather high. Specifically, if an algorithm enjoys a worst-case regret of B with respect to some action, then there must exist another action for which the worst-case regret is at least Ω(nK/B), where n is the horizon and K the number of actions. I also give upper bounds in both the stochastic and adversarial settings showing that this result cannot be improved. For the stochastic case the par…
National University of Singapore（新加坡国立大学）；California Institute of Technology（加州理工学院）
We study the regret of Thompson sampling (TS) algorithms for exponential family bandits, where the reward distribution is from a one-dimensional exponential family, which covers many common reward distributions including Bernoulli, Gaussian, Gamma, Exponential, etc. We propose a Thompson sampling algorithm, termed ExpTS, which uses a novel sampling distribution to avoid the under-estimation of the optimal arm. We provide a tight regret analysis for ExpTS, which simultaneously yields both the finite-time regret bound as well as the asymptotic regret bound. In particular, for a $K$-armed bandit …
Microsoft Research（微软研究院）；University of British Columbia（不列颠哥伦比亚大学）
We consider two agents playing simultaneously the same stochastic three-armed bandit problem. The two agents are cooperating but they cannot communicate. We propose a strategy with no collisions at all between the players (with very high probability), and with near-optimal regret $O(\sqrt{T \log(T)})$. We also argue that the extra logarithmic term $\sqrt{\log(T)}$ should be necessary by proving a lower bound for a full information variant of the problem.
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）
