In this paper, we improve the regret bound for online kernel selection under bandit feedback. Previous algorithm enjoys a $O((\Vert f\Vert^2_{\mathcal{H}_i}+1)K^{\frac{1}{3}}T^{\frac{2}{3}})$ expected bound for Lipschitz loss functions. We prove two types of regret bounds improving the previous bound. For smooth loss functions, we propose an algorithm with a $O(U^{\frac{2}{3}}K^{-\frac{1}{3}}(\sum^K_{i=1}L_T(f^\ast_i))^{\frac{2}{3}})$ expected bound where $L_T(f^\ast_i)$ is the cumulative losses of optimal hypothesis in $\mathbb{H}_{i}=\{f\in\mathcal{H}_i:\Vert f\Vert_{\mathcal{H}_i}\leq U\}$. The data-dependent bound keeps the previous worst-case bound and is smaller if most of candidate kernels match well with the data. For Lipschitz loss functions, we propose an algorithm with a $O(U\sqrt{KT}\ln^{\frac{2}{3}}{T})$ expected bound asymptotically improving the previous bound. We apply the two algorithms to online kernel selection with time constraint and prove new regret bounds matching or improving the previous $O(\sqrt{T\ln{K}} +\Vert f\Vert^2_{\mathcal{H}_i}\max\{\sqrt{T},\frac{T}{\sqrt{\mathcal{R}}}\})$ expected bound where $\mathcal{R}$ is the time budget. Finally, we empirically verify our algorithms on online regression and classification tasks.
Google Research（谷歌研究院）；Yale University（耶鲁大学）；McGill University（麦吉尔大学）
We present simple and efficient algorithms for the batched stochastic multi-armed bandit and batched stochastic linear bandit problems. We prove bounds for their expected regrets that improve over the best-known regret bounds for any number of batches. In particular, our algorithms in both settings achieve the optimal expected regrets by using only a logarithmic number of batches. We also study the batched adversarial multi-armed bandit problem for the first time and find the optimal regret, up to logarithmic factors, of any algorithm with predetermined batch sizes.
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.
Center for Advanced Intelligence Project（高级智能研究中心）；RIKEN（理化学研究所）
The dueling bandit is a learning framework wherein the feedback information in the learning process is restricted to a noisy comparison between a pair of actions. In this research, we address a dueling bandit problem based on a cost function over a continuous space. We propose a stochastic mirror descent algorithm and show that the algorithm achieves an $O(\sqrt{T\log T})$-regret bound under strong convexity and smoothness assumptions for the cost function. Subsequently, we clarify the equivalence between regret minimization in dueling bandit and convex optimization for the cost function. More…
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 …
We consider a decentralized multiplayer game, played over $T$ rounds, with a leader-follower hierarchy described by a directed acyclic graph. For each round, the graph structure dictates the order of the players and how players observe the actions of one another. By the end of each round, all players receive a joint bandit-reward based on their joint action that is used to update the player strategies towards the goal of minimizing the joint pseudo-regret. We present a learning algorithm inspired by the single-player multi-armed bandit problem and show that it achieves sub-linear joint pseudo-…
We analyze the minimax regret of the adversarial bandit convex optimization problem. Focusing on the one-dimensional case, we prove that the minimax regret is $\widetildeΘ(\sqrt{T})$ and partially resolve a decade-old open problem. Our analysis is non-constructive, as we do not present a concrete algorithm that attains this regret rate. Instead, we use minimax duality to reduce the problem to a Bayesian setting, where the convex loss functions are drawn from a worst-case distribution, and then we solve the Bayesian version of the problem with a variant of Thompson Sampling. Our analysis featur…
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…
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…
Tongji University（同济大学）；Shanghai Research Institute for Intelligent Autonomous Systems（上海智能自主系统研究院）
This paper considers no-regret learning for repeated continuous-kernel games with lossy bandit feedback. Since it is difficult to give the explicit model of the utility functions in dynamic environments, the players' action can only be learned with bandit feedback. Moreover, because of unreliable communication channels or privacy protection, the bandit feedback may be lost or dropped at random. Therefore, we study the asynchronous online learning strategy of the players to adaptively adjust the next actions for minimizing the long-term regret loss. The paper provides a novel no-regret learning…
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 …
Università degli Studi di Milano（米兰大学）；Politecnico di Milano（米兰理工大学）
We investigate the problem of bandits with expert advice when the experts are fixed and known distributions over the actions. Improving on previous analyses, we show that the regret in this setting is controlled by information-theoretic quantities that measure the similarity between experts. In some natural special cases, this allows us to obtain the first regret bound for EXP4 that can get arbitrarily close to zero if the experts are similar enough. While for a different algorithm, we provide another bound that describes the similarity between the experts in terms of the KL-divergence, and we…
University of British Columbia（不列颠哥伦比亚大学）；Yahoo! Research（雅虎研究院）
This paper analyses the problem of Gaussian process (GP) bandits with deterministic observations. The analysis uses a branch and bound algorithm that is related to the UCB algorithm of (Srinivas et al., 2010). For GPs with Gaussian observation noise, with variance strictly greater than zero, (Srinivas et al., 2010) proved that the regret vanishes at the approximate rate of $O(\frac{1}{\sqrt{t}})$, where t is the number of observations. To complement their result, we attack the deterministic case and attain a much faster exponential convergence rate. Under some regularity assumptions, we show t…
We consider the problem of online combinatorial optimization under semi-bandit feedback, where a learner has to repeatedly pick actions from a combinatorial decision set in order to minimize the total losses associated with its decisions. After making each decision, the learner observes the losses associated with its action, but not other losses. For this problem, there are several learning algorithms that guarantee that the learner's expected regret grows as $\widetilde{O}(\sqrt{T})$ with the number of rounds $T$. In this paper, we propose an algorithm that improves this scaling to $\widetild…
We consider the restless Markov bandit problem, in which the state of each arm evolves according to a Markov process independently of the learner's actions. We suggest an algorithm that after $T$ steps achieves $\tilde{O}(\sqrt{T})$ regret with respect to the best policy that knows the distributions of all arms. No assumptions on the Markov chains are made except that they are irreducible. In addition, we show that index-based policies are necessarily suboptimal for the considered problem.
Technion - Israel Institute of Technology（以色列理工学院）
We revisit the challenge of designing online algorithms for the bandit convex optimization problem (BCO) which are also scalable to high dimensional problems. Hence, we consider algorithms that are \textit{projection-free}, i.e., based on the conditional gradient method whose only access to the feasible decision set, is through a linear optimization oracle (as opposed to other methods which require potentially much more computationally-expensive subprocedures, such as computing Euclidean projections). We present the first such algorithm that attains $O(T^{3/4})$ expected regret using only $O(T…
UC Berkeley（加州大学伯克利分校）；Facebook AI Research（脸书人工智能研究院）
Algorithms for hyperparameter optimization abound, all of which work well under different and often unverifiable assumptions. Motivated by the general challenge of sequentially choosing which algorithm to use, we study the more specific task of choosing among distributions to use for random hyperparameter optimization. This work is naturally framed in the extreme bandit setting, which deals with sequentially choosing which distribution from a collection to sample in order to minimize (maximize) the single best cost (reward). Whereas the distributions in the standard bandit setting are primaril…
We study the problem of expert advice under partial bandit feedback setting and create a sequential minimax optimal algorithm. Our algorithm works with a more general partial monitoring setting, where, in contrast to the classical bandit feedback, the losses can be revealed in an adversarial manner. Our algorithm adopts a universal prediction perspective, whose performance is analyzed with regret against a general expert selection sequence. The regret we study is against a general competition class that covers many settings (such as the switching or contextual experts settings) and the expert …
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 …
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 …
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.
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…
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.
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…
Microsoft Research（微软研究院）；Princeton University（普林斯顿大学）
Regret bounds in online learning compare the player's performance to $L^*$, the optimal performance in hindsight with a fixed strategy. Typically such bounds scale with the square root of the time horizon $T$. The more refined concept of first-order regret bound replaces this with a scaling $\sqrt{L^*}$, which may be much smaller than $\sqrt{T}$. It is well known that minor variants of standard algorithms satisfy first-order regret bounds in the full information and multi-armed bandit settings. In a COLT 2017 open problem, Agarwal, Krishnamurthy, Langford, Luo, and Schapire raised the issue th…
We study the problem of representation learning in stochastic contextual linear bandits. While the primary concern in this domain is usually to find realizable representations (i.e., those that allow predicting the reward function at any context-action pair exactly), it has been recently shown that representations with certain spectral properties (called HLS) may be more effective for the exploration-exploitation task, enabling LinUCB to achieve constant (i.e., horizon-independent) regret. In this paper, we propose BanditSRL, a representation learning algorithm that combines a novel constraine…
KTH, The Royal Institute of Technology（瑞典皇家理工学院）；Supelec（高等电力学院）；INRIA（法国国家信息与自动化研究所）
We consider stochastic multi-armed bandit problems where the expected reward is a Lipschitz function of the arm, and where the set of arms is either discrete or continuous. For discrete Lipschitz bandits, we derive asymptotic problem specific lower bounds for the regret satisfied by any algorithm, and propose OSLB and CKL-UCB, two algorithms that efficiently exploit the Lipschitz structure of the problem. In fact, we prove that OSLB is asymptotically optimal, as its asymptotic regret matches the lower bound. The regret analysis of our algorithms relies on a new concentration inequality for wei…
Google（谷歌）；Google Research（谷歌研究院）；DeepMind（深度思维）；University of Alberta（阿尔伯塔大学）
We propose ${\tt AdaTS}$, a Thompson sampling algorithm that adapts sequentially to bandit tasks that it interacts with. The key idea in ${\tt AdaTS}$ is to adapt to an unknown task prior distribution by maintaining a distribution over its parameters. When solving a bandit task, that uncertainty is marginalized out and properly accounted for. ${\tt AdaTS}$ is a fully-Bayesian algorithm that can be implemented efficiently in several classes of bandit problems. We derive upper bounds on its Bayes regret that quantify the loss due to not knowing the task prior, and show that it is small. Our theo…
We study an extension of standard bandit problem in which there are R layers of experts. Multi-layered experts make selections layer by layer and only the experts in the last layer can play arms. The goal of the learning policy is to minimize the total regret in this hierarchical experts setting. We first analyze the case that total regret grows linearly with the number of layers. Then we focus on the case that all experts are playing Upper Confidence Bound (UCB) strategy and give several sub-linear upper bounds for different circumstances. Finally, we design some experiments to help the regre…
Optimal Regret for Policy Optimization in Contextual Bandits
School of Computer Science and AI, Tel-Aviv University, Tel-Aviv, Israel（计算机科学与人工智能学院，特拉维夫大学，特拉维夫，以色列）；Google Research, Tel-Aviv, Israel（谷歌研究，特拉维夫，以色列）
University of California, Berkeley（加州大学伯克利分校）；Indian Institute of Science（印度科学院）
We consider the problem of online learning in misspecified linear stochastic multi-armed bandit problems. Regret guarantees for state-of-the-art linear bandit algorithms such as Optimism in the Face of Uncertainty Linear bandit (OFUL) hold under the assumption that the arms expected rewards are perfectly linear in their features. It is, however, of interest to investigate the impact of potential misspecification in linear bandit models, where the expected rewards are perturbed away from the linear subspace determined by the arms features. Although OFUL has recently been shown to be robust to r…
Peking University（北京大学）；Pazhou Lab（琶洲实验室）；University of California, Los Angles（加利福尼亚大学洛杉矶分校）
Despite a large amount of effort in dealing with heavy-tailed error in machine learning, little is known when moments of the error can become non-existential: the random noise $η$ satisfies Pr$\left[|η| > |y|\right] \le 1/|y|^α$ for some $α> 0$. We make the first attempt to actively handle such super heavy-tailed noise in bandit learning problems: We propose a novel robust statistical estimator, mean of medians, which estimates a random variable by computing the empirical mean of a sequence of empirical medians. We then present a generic reductionist algorithmic framework for solving bandit le…
We study the nonstationary stochastic Multi-Armed Bandit (MAB) problem in which the distribution of rewards associated with each arm are assumed to be time-varying and the total variation in the expected rewards is subject to a variation budget. The regret of a policy is defined by the difference in the expected cumulative rewards obtained using the policy and using an oracle that selects the arm with the maximum mean reward at each time. We characterize the performance of the proposed policies in terms of the worst-case regret, which is the supremum of the regret over the set of reward distri…
We study a general multi-dueling bandit problem, where an agent compares multiple options simultaneously and aims to minimize the regret due to selecting suboptimal arms. This setting generalizes the traditional two-dueling bandit problem and finds many real-world applications involving subjective feedback on multiple options. We start with the two-dueling bandit setting and propose two efficient algorithms, DoublerBAI and MultiSBM-Feedback. DoublerBAI provides a generic schema for translating known results on best arm identification algorithms to the dueling bandit problem, and achieves a reg…
We study the optimal batch-regret tradeoff for batch linear contextual bandits. For any batch number $M$, number of actions $K$, time horizon $T$, and dimension $d$, we provide an algorithm and prove its regret guarantee, which, due to technical reasons, features a two-phase expression as the time horizon $T$ grows. We also prove a lower bound theorem that surprisingly shows the optimality of our two-phase regret upper bound (up to logarithmic factors) in the \emph{full range} of the problem parameters, therefore establishing the exact batch-regret tradeoff. Compared to the recent work \citep{…
California Institute of Technology（加州理工学院）；University of Pennsylvania（宾夕法尼亚大学）；Saarland University（萨尔大学）
Many applications require optimizing an unknown, noisy function that is expensive to evaluate. We formalize this task as a multi-armed bandit problem, where the payoff function is either sampled from a Gaussian process (GP) or has low RKHS norm. We resolve the important open problem of deriving regret bounds for this setting, which imply novel convergence rates for GP optimization. We analyze GP-UCB, an intuitive upper-confidence based algorithm, and bound its cumulative regret in terms of maximal information gain, establishing a novel connection between GP optimization and experimental design…
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…
We study the attainable regret for online linear optimization problems with bandit feedback, where unlike the full-information setting, the player can only observe its own loss rather than the full loss vector. We show that the price of bandit information in this setting can be as large as $d$, disproving the well-known conjecture that the regret for bandit linear optimization is at most $\sqrt{d}$ times the full-information regret. Surprisingly, this is shown using "trivial" modifications of standard domains, which have no effect in the full-information setting. This and other results we pres…
Politecnico di Milano（米兰理工大学）；Facebook AI Research（Facebook AI研究院）
The linear contextual bandit literature is mostly focused on the design of efficient learning algorithms for a given representation. However, a contextual bandit problem may admit multiple linear representations, each one with different characteristics that directly impact the regret of the learning algorithm. In particular, recent works showed that there exist "good" representations for which constant problem-dependent regret can be achieved. In this paper, we first provide a systematic analysis of the different definitions of "good" representations proposed in the literature. We then propose…
Microsoft Research（微软研究院）；Princeton University（普林斯顿大学）
We give an oracle-based algorithm for the adversarial contextual bandit problem, where either contexts are drawn i.i.d. or the sequence of contexts is known a priori, but where the losses are picked adversarially. Our algorithm is computationally efficient, assuming access to an offline optimization oracle, and enjoys a regret of order $O((KT)^{\frac{2}{3}}(\log N)^{\frac{1}{3}})$, where $K$ is the number of actions, $T$ is the number of iterations and $N$ is the number of baseline policies. Our result is the first to break the $O(T^{\frac{3}{4}})$ barrier that is achieved by recently introduc…
Learning in Markovian bandits with non-observable states and constrained decision epochs
IRIT, Université de Toulouse, CNRS, Toulouse INP（图卢兹大学IRIT实验室，法国国家科学研究中心，图卢兹国立理工学院）；Ikerbasque-UPV/EHU, University of the Basque Country（巴斯克科学与技术联盟-巴斯克大学）
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…
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…
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.
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.
