我们提出了一个新的强盗算法，SAO（随机和对抗最优），其遗憾，本质上是最优的对抗性奖励和随机奖励。具体地，SAO结合了Exp3的平方根最坏情况遗憾（Auer等人，SIAM J. on Computing 2002）和UCB 1的（多）对数遗憾（Auer et al.，机器学习2002）的随机奖励。对抗性奖励和随机奖励是（非贝叶斯）多臂强盗文献中的两个主要设置。之前关于多武装土匪的工作将它们分开对待，并且不试图联合优化两者。我们的结果属于一个一般的主题，实现良好的最坏情况下的性能，同时也利用了“好”的问题的情况下，一个重要的问题，在设计的算法与部分已知的输入。
英文摘要：
We present a new bandit algorithm, SAO (Stochastic and Adversarial Optimal), whose regret is, essentially, optimal both for adversarial rewards and for stochastic rewards. Specifically, SAO combines the square-root worst-case regret of Exp3 (Auer et al., SIAM J. on Computing 2002) and the (poly)logarithmic regret of UCB1 (Auer et al., Machine Learning 2002) for stochastic rewards. Adversarial rewards and stochastic rewards are the two main settings in the literature on (non-Bayesian) multi-armed bandits. Prior work on multi-armed bandits treats them separately, and does not attempt to jointly optimize for both. Our result falls into a general theme of achieving good worst-case performance while also taking advantage of "nice" problem instances, an important issue in the design of algorithms with partially known inputs.
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 …
We derive improved regret bounds for the Tsallis-INF algorithm of Zimmert and Seldin (2021). We show that in adversarial regimes with a $(Δ,C,T)$ self-bounding constraint the algorithm achieves $\mathcal{O}\left(\left(\sum_{i\neq i^*} \frac{1}{Δ_i}\right)\log_+\left(\frac{(K-1)T}{\left(\sum_{i\neq i^*} \frac{1}{Δ_i}\right)^2}\right)+\sqrt{C\left(\sum_{i\neq i^*}\frac{1}{Δ_i}\right)\log_+\left(\frac{(K-1)T}{C\sum_{i\neq i^*}\frac{1}{Δ_i}}\right)}\right)$ regret bound, where $T$ is the time horizon, $K$ is the number of arms, $Δ_i$ are the suboptimality gaps, $i^*$ is the best arm, $C$ is the co…
We study the stochastic multi-armed bandits problem in the presence of adversarial corruption. We present a new algorithm for this problem whose regret is nearly optimal, substantially improving upon previous work. Our algorithm is agnostic to the level of adversarial contamination and can tolerate a significant amount of corruption with virtually no degradation in performance.
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.
New York University（纽约大学）；Stanford University（斯坦福大学）；University of Florida（佛罗里达大学）
In this paper we study the adversarial combinatorial bandit with a known non-linear reward function, extending existing work on adversarial linear combinatorial bandit. {The adversarial combinatorial bandit with general non-linear reward is an important open problem in bandit literature, and it is still unclear whether there is a significant gap from the case of linear reward, stochastic bandit, or semi-bandit feedback.} We show that, with $N$ arms and subsets of $K$ arms being chosen at each of $T$ time periods, the minimax optimal regret is $\widetildeΘ_{d}(\sqrt{N^d T})$ if the reward funct…
Bridging Adversarial and Nonstationary Multi-armed Bandit
Rotman School of Management, University of Toronto（多伦多大学罗特曼管理学院）；The Hong Kong University of Science and Technology（香港科技大学）；School of Data Science, The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳）数据科学学院）
University of Copenhagen（哥本哈根大学）；University of Southern California（南加州大学）
We develop the first general semi-bandit algorithm that simultaneously achieves $\mathcal{O}(\log T)$ regret for stochastic environments and $\mathcal{O}(\sqrt{T})$ regret for adversarial environments without knowledge of the regime or the number of rounds $T$. The leading problem-dependent constants of our bounds are not only optimal in some worst-case sense studied previously, but also optimal for two concrete instances of semi-bandit problems. Our algorithm and analysis extend the recent work of (Zimmert & Seldin, 2019) for the special case of multi-armed bandit, but importantly requires a …
Microsoft Research NYC（微软研究院纽约分院）；Cornell University（康奈尔大学）；Google Research NYC（谷歌研究院纽约分院）
We study "adversarial scaling", a multi-armed bandit model where rewards have a stochastic and an adversarial component. Our model captures display advertising where the "click-through-rate" can be decomposed to a (fixed across time) arm-quality component and a non-stochastic user-relevance component (fixed across arms). Despite the relative stochasticity of our model, we demonstrate two settings where most bandit algorithms suffer. On the positive side, we show that two algorithms, one from the action elimination and one from the mirror descent family are adaptive enough to be robust to adver…
Microsoft Research（微软研究院）；Facebook（脸书公司）；University of Maryland（马里兰大学）
We consider Bandits with Knapsacks (henceforth, BwK), a general model for multi-armed bandits under supply/budget constraints. In particular, a bandit algorithm needs to solve a well-known knapsack problem: find an optimal packing of items into a limited-size knapsack. The BwK problem is a common generalization of numerous motivating examples, which range from dynamic pricing to repeated auctions to dynamic ad allocation to network routing and scheduling. While the prior work on BwK focused on the stochastic version, we pioneer the other extreme in which the outcomes can be chosen adversariall…
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）
University of Copenhagen（哥本哈根大学）；ICREA（加泰罗尼亚高等研究院）；Pompeu Fabra University（庞培法布拉大学）
We present a new strategy for gap estimation in randomized algorithms for multiarmed bandits and combine it with the EXP3++ algorithm of Seldin and Slivkins (2014). In the stochastic regime the strategy reduces dependence of regret on a time horizon from $(\ln t)^3$ to $(\ln t)^2$ and eliminates an additive factor of order $Δe^{1/Δ^2}$, where $Δ$ is the minimal gap of a problem instance. In the adversarial regime regret guarantee remains unchanged.
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…
We study the K-armed dueling bandit problem which is a variation of the classical Multi-Armed Bandit (MAB) problem in which the learner receives only relative feedback about the selected pairs of arms. We propose a new algorithm called Relative Exponential-weight algorithm for Exploration and Exploitation (REX3) to handle the adversarial utility-based formulation of this problem. This algorithm is a non-trivial extension of the Exponential-weight algorithm for Exploration and Exploitation (EXP3) algorithm. We prove a finite time expected regret upper bound of order O(sqrt(K ln(K)T)) for this a…
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.
University of California, Davis（加州大学戴维斯分校）；University of California, Los Angeles（加州大学洛杉矶分校）；Amazon（亚马逊）
Stochastic linear contextual bandit algorithms have substantial applications in practice, such as recommender systems, online advertising, clinical trials, etc. Recent works show that optimal bandit algorithms are vulnerable to adversarial attacks and can fail completely in the presence of attacks. Existing robust bandit algorithms only work for the non-contextual setting under the attack of rewards and cannot improve the robustness in the general and popular contextual bandit environment. In addition, none of the existing methods can defend against attacked context. In this work, we provide t…
Bilkent University（比尔肯特大学）；Massachusetts Institute of Technology（麻省理工学院）
We investigate the adversarial bandit problem with multiple plays under semi-bandit feedback. We introduce a highly efficient algorithm that asymptotically achieves the performance of the best switching $m$-arm strategy with minimax optimal regret bounds. To construct our algorithm, we introduce a new expert advice algorithm for the multiple-play setting. By using our expert advice algorithm, we additionally improve the best-known high-probability bound for the multi-play setting by $O(\sqrt{m})$. Our results are guaranteed to hold in an individual sequence manner since we have no statistical …
University of Southern California（南加州大学）；The Chinese University of Hong Kong（香港中文大学）
In this work, we develop linear bandit algorithms that automatically adapt to different environments. By plugging a novel loss estimator into the optimization problem that characterizes the instance-optimal strategy, our first algorithm not only achieves nearly instance-optimal regret in stochastic environments, but also works in corrupted environments with additional regret being the amount of corruption, while the state-of-the-art (Li et al., 2019) achieves neither instance-optimality nor the optimal dependence on the corruption amount. Moreover, by equipping this algorithm with an adversari…
The classic cake-cutting problem provides a model for addressing the fair and efficient allocation of a divisible, heterogeneous resource among agents with distinct preferences. Focusing on a standard formulation of cake cutting, in which each agent must receive a contiguous piece of the cake in an offline setting, this work instead focuses on online allocating non-contiguous pieces of cake among agents and establishes algorithmic results for fairness measures. In this regard, we made use of classic adversarial multi-armed bandits to achieve sub-linear Fairness and Revenue Regret at the same t…
We derive an algorithm that achieves the optimal (within constants) pseudo-regret in both adversarial and stochastic multi-armed bandits without prior knowledge of the regime and time horizon. The algorithm is based on online mirror descent (OMD) with Tsallis entropy regularization with power $α=1/2$ and reduced-variance loss estimators. More generally, we define an adversarial regime with a self-bounding constraint, which includes stochastic regime, stochastically constrained adversarial regime (Wei and Luo), and stochastic regime with adversarial corruptions (Lykouris et al.) as special case…
Indian Institute of Science（印度科学研究所）；Tel Aviv University（特拉维夫大学）；Google Tel Aviv（谷歌特拉维夫）
We introduce the problem of regret minimization in Adversarial Dueling Bandits. As in classic Dueling Bandits, the learner has to repeatedly choose a pair of items and observe only a relative binary `win-loss' feedback for this pair, but here this feedback is generated from an arbitrary preference matrix, possibly chosen adversarially. Our main result is an algorithm whose $T$-round regret compared to the \emph{Borda-winner} from a set of $K$ items is $\tilde{O}(K^{1/3}T^{2/3})$, as well as a matching $Ω(K^{1/3}T^{2/3})$ lower bound. We also prove a similar high probability regret bound. We fu…
We investigate the optimality of perturbation based algorithms in the stochastic and adversarial multi-armed bandit problems. For the stochastic case, we provide a unified regret analysis for both sub-Weibull and bounded perturbations when rewards are sub-Gaussian. Our bounds are instance optimal for sub-Weibull perturbations with parameter 2 that also have a matching lower tail bound, and all bounded support perturbations where there is sufficient probability mass at the extremes of the support. For the adversarial setting, we prove rigorous barriers against two natural solution approaches us…
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…
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）
Gatsby Computational Neuroscience Unit（盖茨比计算神经科学单元）；University College London（伦敦大学学院）；DeepMind（深度思维）
We make three contributions to the theory of k-armed adversarial bandits. First, we prove a first-order bound for a modified variant of the INF strategy by Audibert and Bubeck [2009], without sacrificing worst case optimality or modifying the loss estimators. Second, we provide a variance analysis for algorithms based on follow the regularised leader, showing that without adaptation the variance of the regret is typically Ω(n^2) where n is the horizon. Finally, we study bounds that depend on the degree of separation of the arms, generalising the results by Cowan and Katehakis [2015] from the s…
Microsoft Research（微软研究院）；Laboratoire Jean Kuntzmann（让·孔茨曼实验室）
We introduce the problem of sleeping dueling bandits with stochastic preferences and adversarial availabilities (DB-SPAA). In almost all dueling bandit applications, the decision space often changes over time; eg, retail store management, online shopping, restaurant recommendation, search engine optimization, etc. Surprisingly, this `sleeping aspect' of dueling bandits has never been studied in the literature. Like dueling bandits, the goal is to compete with the best arm by sequentially querying the preference feedback of item pairs. The non-triviality however results due to the non-stationar…
We introduce a new model of stochastic bandits with adversarial corruptions which aims to capture settings where most of the input follows a stochastic pattern but some fraction of it can be adversarially changed to trick the algorithm, e.g., click fraud, fake reviews and email spam. The goal of this model is to encourage the design of bandit algorithms that (i) work well in mixed adversarial and stochastic models, and (ii) whose performance deteriorates gracefully as we move from fully stochastic to fully adversarial models. In our model, the rewards for all arms are initially drawn from a di…
Adversarial bandit optimization for approximately linear functions
Joint Graduate School of Mathematics for Innovation, Kyushu University, Japan（创新数学联合研究生院，九州大学，日本）；Department of Informatics, Kyushu University, Japan（信息学系，九州大学，日本）；RIKEN AIP, Japan（日本理化学研究所AIP）
In this work we consider adversarial contextual bandits with risk constraints. At each round, nature prepares a context, a cost for each arm, and additionally a risk for each arm. The learner leverages the context to pull an arm and then receives the corresponding cost and risk associated with the pulled arm. In addition to minimizing the cumulative cost, the learner also needs to satisfy long-term risk constraints -- the average of the cumulative risk from all pulled arms should not be larger than a pre-defined threshold. To address this problem, we first study the full information setting wh…
Department of Computer Science, ETH Zurich（苏黎世联邦理工学院计算机科学系）
We consider a setting where multiple players sequentially choose among a common set of actions (arms). Motivated by a cognitive radio networks application, we assume that players incur a loss upon colliding, and that communication between players is not possible. Existing approaches assume that the system is stationary. Yet this assumption is often violated in practice, e.g., due to signal strength fluctuations. In this work, we design the first Multi-player Bandit algorithm that provably works in arbitrarily changing environments, where the losses of the arms may even be chosen by an adversar…
We study the adversarial multi-armed bandit problem and create a completely online algorithmic framework that is invariant under arbitrary translations and scales of the arm losses. We study the expected performance of our algorithm against a generic competition class, 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 arm selection sequences, which is the difference between our losses and a competing loss sequence. The competition class can be d…
University California San Diego（加州大学圣地亚哥分校）；Shanghai Jiao Tong University（上海交通大学）；Nanjing University of Posts and Telecommunications（南京邮电大学）
We study the problem of stochastic bandits with adversarial corruptions in the cooperative multi-agent setting, where $V$ agents interact with a common $K$-armed bandit problem, and each pair of agents can communicate with each other to expedite the learning process. In the problem, the rewards are independently sampled from distributions across all agents and rounds, but they may be corrupted by an adversary. Our goal is to minimize both the overall regret and communication cost across all agents. We first show that an additive term of corruption is unavoidable for any algorithm in this probl…
Contextual bandit algorithms are applied in a wide range of domains, from advertising to recommender systems, from clinical trials to education. In many of these domains, malicious agents may have incentives to attack the bandit algorithm to induce it to perform a desired behavior. For instance, an unscrupulous ad publisher may try to increase their own revenue at the expense of the advertisers; a seller may want to increase the exposure of their products, or thwart a competitor's advertising campaign. In this paper, we study several attack scenarios and show that a malicious agent can force a…
Bandit Multi-linear DR-Submodular Maximization and Its Applications on Adversarial Submodular Bandits
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）；Microsoft Research（微软研究院）；Fuzhou University（福州大学）
We develop a new approach to obtaining high probability regret bounds for online learning with bandit feedback against an adaptive adversary. While existing approaches all require carefully constructing optimistic and biased loss estimators, our approach uses standard unbiased estimators and relies on a simple increasing learning rate schedule, together with the help of logarithmically homogeneous self-concordant barriers and a strengthened Freedman's inequality. Besides its simplicity, our approach enjoys several advantages. First, the obtained high-probability regret bounds are data-dependen…
We consider the problem of falsifying safety requirements of Cyber-Physical Systems expressed in signal temporal logic (STL). This problem can be turned into an optimization problem via STL robustness functions. In this paper, our focus is in falsifying systems with multiple requirements. We propose to solve such conjunctive requirements using online generative adversarial networks (GANs) as test generators. Our main contribution is an algorithm which falsifies a conjunctive requirement $φ_1 \land \cdots \land φ_n$ by using a GAN for each requirement $φ_i$ separately. Using ideas from multi-ar…
