Many past attempts at modeling repeated Cournot games assume that demand is stationary. This does not align with real-world scenarios in which market demands can evolve over a product's lifetime for a myriad of reasons. In this paper, we model repeated Cournot games with non-stationary demand such that firms/agents face separate instances of non-stationary multi-armed bandit problem. The set of arms/actions that an agent can choose from represents discrete production quantities; here, the action space is ordered. Agents are independent and autonomous, and cannot observe anything from the environment; they can only see their own rewards after taking an action, and only work towards maximizing these rewards. We propose a novel algorithm 'Adaptive with Weighted Exploration (AWE) $ε$-greedy' which is remotely based on the well-known $ε$-greedy approach. This algorithm detects and quantifies changes in rewards due to varying market demand and varies learning rate and exploration rate in proportion to the degree of changes in demand, thus enabling agents to better identify new optimal actions. For efficient exploration, it also deploys a mechanism for weighing actions that takes advantage of the ordered action space. We use simulations to study the emergence of various equilibria in the market. In addition, we study the scalability of our approach in terms number of total agents in the system and the size of action space. We consider both symmetric and asymmetric firms in our models. We found that using our proposed method, agents are able to swiftly change their course of action according to the changes in demand, and they also engage in collusive behavior in many simulations.
We investigate the use of a multi-agent multi-armed bandit (MA-MAB) setting for modeling repeated Cournot oligopoly games, where the firms acting as agents choose from the set of arms representing production quantity (a discrete value). Agents interact with separate and independent bandit problems. In this formulation, each agent makes sequential choices among arms to maximize its own reward. Agents do not have any information about the environment; they can only see their own rewards after taking an action. However, the market demand is a stationary function of total industry output, and rand…
Crowdsourcing markets have emerged as a popular platform for matching available workers with tasks to complete. The payment for a particular task is typically set by the task's requester, and may be adjusted based on the quality of the completed work, for example, through the use of "bonus" payments. In this paper, we study the requester's problem of dynamically adjusting quality-contingent payments for tasks. We consider a multi-round version of the well-known principal-agent model, whereby in each round a worker makes a strategic choice of the effort level which is not directly observable by…
KTH Royal Institute of Technology（KTH皇家理工学院）；Digital Futures（数字未来中心）；Duke University（杜克大学）；Air Force Research Laboratory（美国空军研究实验室）
We consider risk-averse learning in repeated unknown games where the goal of the agents is to minimize their individual risk of incurring significantly high cost. Specifically, the agents use the conditional value at risk (CVaR) as a risk measure and rely on bandit feedback in the form of the cost values of the selected actions at every episode to estimate their CVaR values and update their actions. A major challenge in using bandit feedback to estimate CVaR is that the agents can only access their own cost values, which, however, depend on the actions of all agents. To address this challenge,…
The Technion - Israel Institute of Technology（以色列理工学院）
In many embedded systems, such as imaging sys- tems, the system has a single designated purpose, and same threads are executed repeatedly. Profiling thread behavior, allows the system to allocate each thread its resources in a way that improves overall system performance. We study an online resource al- locationproblem,wherearesourcemanagersimulta- neously allocates resources (exploration), learns the impact on the different consumers (learning) and im- proves allocation towards optimal performance (ex- ploitation). We build on the rich framework of multi- armed bandits and present online and …
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…
Q learning is widely used to simulate the behaviors of generation companies (GenCos) in an electricity market. However, existing Q learning method usually requires numerous iterations to converge, which is time-consuming and inefficient in practice. To enhance the calculation efficiency, a novel Q learning algorithm improved by dichotomy is proposed in this paper. This method modifies the update process of the Q table by dichotomizing the state space and the action space step by step. Simulation results in a repeated Cournot game show the effectiveness of the proposed algorithm.
Psychological research shows that enjoyment of many goods is subject to satiation, with short-term satisfaction declining after repeated exposures to the same item. Nevertheless, proposed algorithms for powering recommender systems seldom model these dynamics, instead proceeding as though user preferences were fixed in time. In this work, we introduce rebounding bandits, a multi-armed bandit setup, where satiation dynamics are modeled as time-invariant linear dynamical systems. Expected rewards for each arm decline monotonically with consecutive exposures to it and rebound towards the initial …
Microsoft Research（微软研究院）；Columbia University（哥伦比亚大学）；Yahoo! Labs（雅虎研究院）；Princeton University（普林斯顿大学）
We present a new algorithm for the contextual bandit learning problem, where the learner repeatedly takes one of $K$ actions in response to the observed context, and observes the reward only for that chosen action. Our method assumes access to an oracle for solving fully supervised cost-sensitive classification problems and achieves the statistically optimal regret guarantee with only $\tilde{O}(\sqrt{KT/\log N})$ oracle calls across all $T$ rounds, where $N$ is the number of policies in the policy class we compete against. By doing so, we obtain the most practical contextual bandit learning a…
We study the sequential resource allocation problem where a decision maker repeatedly allocates budgets between resources. Motivating examples include allocating limited computing time or wireless spectrum bands to multiple users (i.e., resources). At each timestep, the decision maker should distribute its available budgets among different resources to maximize the expected reward, or equivalently to minimize the cumulative regret. In doing so, the decision maker should learn the value of the resources allocated for each user from feedback on each user's received reward. For example, users may…
University of California, San Diego（加州大学圣地亚哥分校）；University of Southampton（南安普顿大学）
This paper investigates the adversarial Bandits with Knapsack (BwK) online learning problem, where a player repeatedly chooses to perform an action, pays the corresponding cost, and receives a reward associated with the action. The player is constrained by the maximum budget $B$ that can be spent to perform actions, and the rewards and the costs of the actions are assigned by an adversary. This problem has only been studied in the restricted setting where the reward of an action is greater than the cost of the action, while we provide a solution in the general setting. Namely, we propose EXP3.…
We address the problem of learning in an online setting where the learner repeatedly observes features, selects among a set of actions, and receives reward for the action taken. We provide the first efficient algorithm with an optimal regret. Our algorithm uses a cost sensitive classification learner as an oracle and has a running time $\mathrm{polylog}(N)$, where $N$ is the number of classification rules among which the oracle might choose. This is exponentially faster than all previous algorithms that achieve optimal regret in this setting. Our formulation also enables us to create an algori…
Contextual bandit learning is a reinforcement learning problem where the learner repeatedly receives a set of features (context), takes an action and receives a reward based on the action and context. We consider this problem under a realizability assumption: there exists a function in a (known) function class, always capable of predicting the expected reward, given the action and context. Under this assumption, we show three things. We present a new algorithm---Regressor Elimination--- with a regret similar to the agnostic setting (i.e. in the absence of realizability assumption). We prove a …
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…
Paderborn University（帕德博恩大学）；University of Munich（慕尼黑大学）
We study the non-stationary dueling bandits problem with $K$ arms, where the time horizon $T$ consists of $M$ stationary segments, each of which is associated with its own preference matrix. The learner repeatedly selects a pair of arms and observes a binary preference between them as feedback. To minimize the accumulated regret, the learner needs to pick the Condorcet winner of each stationary segment as often as possible, despite preference matrices and segment lengths being unknown. We propose the $\mathrm{Beat\, the\, Winner\, Reset}$ algorithm and prove a bound on its expected binary weak…
Incentivized Learning in Principal-Agent Bandit Games
Centre de Mathématiques Appliquées（应用数学中心）；CNRS（法国国家科学研究中心）；École polytechnique（巴黎综合理工学院）；Institut Polytechnique de Paris（巴黎理工学院）；Université Paris-Saclay（巴黎萨克雷大学）；INRIA（法国国家信息与自动化研究所）；University of California, Berkeley（加州大学伯克利分校）；Ecole Normale Supérieure（高等师范学院）；PSL Research University（巴黎文理研究大学）
IBM Research（IBM研究院）；Yahoo! Research（雅虎研究院）；Georgia Institute of Technology（佐治亚理工学院）；Princeton University（普林斯顿大学）
We address the problem of learning in an online, bandit setting where the learner must repeatedly select among $K$ actions, but only receives partial feedback based on its choices. We establish two new facts: First, using a new algorithm called Exp4.P, we show that it is possible to compete with the best in a set of $N$ experts with probability $1-δ$ while incurring regret at most $O(\sqrt{KT\ln(N/δ)})$ over $T$ time steps. The new algorithm is tested empirically in a large-scale, real-world dataset. Second, we give a new algorithm called VE that competes with a possibly infinite set of polici…
Developing efficient sequential bidding strategies for repeated auctions is an important practical challenge in various marketing tasks. In this setting, the bidding agent obtains information, on both the value of the item at sale and the behavior of the other bidders, only when she wins the auction. Standard bandit theory does not apply to this problem due to the presence of action-dependent censoring. In this work, we consider second-price auctions and propose novel, efficient UCB-like algorithms for this task. These algorithms are analyzed in the stochastic setting, assuming regularity of t…
Institute of Science Tokyo（东京科学大学）；University of Tokyo（东京大学）
In this paper, we study a new decision-making problem called the bandit max-min fair allocation (BMMFA) problem. The goal of this problem is to maximize the minimum utility among agents with additive valuations by repeatedly assigning indivisible goods to them. One key feature of this problem is that each agent's valuation for each item can only be observed through the semi-bandit feedback, while existing work supposes that the item values are provided at the beginning of each round. Another key feature is that the algorithm's reward function is not additive with respect to rounds, unlike most…
The University of Sydney Business School（悉尼大学商学院）；University of Southampton（南安普顿大学）
In budget-limited multi-armed bandit (MAB) problems, the learner's actions are costly and constrained by a fixed budget. Consequently, an optimal exploitation policy may not be to pull the optimal arm repeatedly, as is the case in other variants of MAB, but rather to pull the sequence of different arms that maximises the agent's total reward within the budget. This difference from existing MABs means that new approaches to maximising the total reward are required. Given this, we develop two pulling policies, namely: (i) KUBE; and (ii) fractional KUBE. Whereas the former provides better perform…
Multi-armed bandit problems provide a framework to identify the optimal intervention over a sequence of repeated experiments. Without additional assumptions, minimax optimal performance (measured by cumulative regret) is well-understood. With access to additional observed variables that d-separate the intervention from the outcome (i.e., they are a d-separator), recent "causal bandit" algorithms provably incur less regret. However, in practice it is desirable to be agnostic to whether observed variables are a d-separator. Ideally, an algorithm should be adaptive; that is, perform nearly as wel…
Meta AI；Univ. Lille（里尔大学）；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）；Centrale Lille（里尔中央理工学院）
Elimination algorithms for bandit identification, which prune the plausible correct answers sequentially until only one remains, are computationally convenient since they reduce the problem size over time. However, existing elimination strategies are often not fully adaptive (they update their sampling rule infrequently) and are not easy to extend to combinatorial settings, where the set of answers is exponentially large in the problem dimension. On the other hand, most existing fully-adaptive strategies to tackle general identification problems are computationally demanding since they repeate…
Massachusetts Institute of Technology（麻省理工学院）；Université Paris Diderot（巴黎第七大学）
Motivated by online advertising auctions, we consider repeated Vickrey auctions where goods of unknown value are sold sequentially and bidders only learn (potentially noisy) information about a good's value once it is purchased. We adopt an online learning approach with bandit feedback to model this problem and derive bidding strategies for two models: stochastic and adversarial. In the stochastic model, the observed values of the goods are random variables centered around the true value of the good. In this case, logarithmic regret is achievable when competing against well behaved adversaries…
We propose an adaptive sampling approach for multiple testing which aims to maximize statistical power while ensuring anytime false discovery control. We consider $n$ distributions whose means are partitioned by whether they are below or equal to a baseline (nulls), versus above the baseline (actual positives). In addition, each distribution can be sequentially and repeatedly sampled. Inspired by the multi-armed bandit literature, we provide an algorithm that takes as few samples as possible to exceed a target true positive proportion (i.e. proportion of actual positives discovered) while givi…
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…
Masdar Institute of Science and Technology（马斯达尔科学技术研究院）
In repeated stochastic games (RSGs), an agent must quickly adapt to the behavior of previously unknown associates, who may themselves be learning. This machine-learning problem is particularly challenging due, in part, to the presence of multiple (even infinite) equilibria and inherently large strategy spaces. In this paper, we introduce a method to reduce the strategy space of two-player general-sum RSGs to a handful of expert strategies. This process, called Mega, effectually reduces an RSG to a bandit problem. We show that the resulting strategy space preserves several important properties …
We consider combinatorial semi-bandits over a set of arms ${\cal X} \subset \{0,1\}^d$ where rewards are uncorrelated across items. For this problem, the algorithm ESCB yields the smallest known regret bound $R(T) = {\cal O}\Big( {d (\ln m)^2 (\ln T) \over Δ_{\min} }\Big)$, but it has computational complexity ${\cal O}(|{\cal X}|)$ which is typically exponential in $d$, and cannot be used in large dimensions. We propose the first algorithm which is both computationally and statistically efficient for this problem with regret $R(T) = {\cal O} \Big({d (\ln m)^2 (\ln T)\over Δ_{\min} }\Big)$ and …
Tel Aviv University（特拉维夫大学）；Institute for Advanced Study（高等研究院）；Microsoft Research（微软研究院）；Università degli Studi di Milano（米兰大学）；University of Insubria（因苏布里亚大学）；Technion（以色列理工学院）；Weizmann Institute of Science（魏茨曼科学研究所）
We present and study a partial-information model of online learning, where a decision maker repeatedly chooses from a finite set of actions, and observes some subset of the associated losses. This naturally models several situations where the losses of different actions are related, and knowing the loss of one action provides information on the loss of other actions. Moreover, it generalizes and interpolates between the well studied full-information setting (where all losses are revealed) and the bandit setting (where only the loss of the action chosen by the player is revealed). We provide se…
Preference-centric Bandits: Optimality of Mixtures and Regret-efficient Algorithms
Rensselaer Polytechnic Institute（伦斯勒理工学院）；Imperial College London（伦敦帝国学院）；Indian Institute of Technology Madras（马德拉斯印度理工学院）；Google Research India（谷歌印度研究院）
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…
Institute for Interdisciplinary Information Sciences, Tsinghua University（清华大学交叉信息研究院）；Microsoft Research（微软研究院）
In this paper, we study the stochastic version of the one-sided full information bandit problem, where we have $K$ arms $[K] = \{1, 2, \ldots, K\}$, and playing arm $i$ would gain reward from an unknown distribution for arm $i$ while obtaining reward feedback for all arms $j \ge i$. One-sided full information bandit can model the online repeated second-price auctions, where the auctioneer could select the reserved price in each round and the bidders only reveal their bids when their bids are higher than the reserved price. In this paper, we present an elimination-based algorithm to solve the p…
Although the classical version of the Multi-Armed Bandits (MAB) framework has been applied successfully to several practical problems, in many real-world applications, the possible actions are not presented to the learner simultaneously, such as in the Internet campaign management and environmental monitoring settings. Instead, in such applications, a set of options is presented sequentially to the learner within a time span, and this process is repeated throughout a time horizon. At each time, the learner is asked whether to select the proposed option or not. We define this scenario as the Se…
In this paper, we study the multi-objective bandits (MOB) problem, where a learner repeatedly selects one arm to play and then receives a reward vector consisting of multiple objectives. MOB has found many real-world applications as varied as online recommendation and network routing. On the other hand, these applications typically contain contextual information that can guide the learning process which, however, is ignored by most of existing work. To utilize this information, we associate each arm with a context vector and assume the reward follows the generalized linear model (GLM). We adop…
Beyond Static Bias: Adaptive Multi-Fidelity Bandits with Improving Proxies
Department of Industrial and Systems Engineering, University of Houston（休斯敦大学工业与系统工程系）；School of Electrical Engineering and Computer Science, Oregon State University（俄勒冈州立大学电气工程与计算机科学学院）
The American University in Cairo（开罗美国大学）；Université du Québec à Montréal（魁北克大学蒙特利尔分校）
This paper designs a sequential repeated game of a micro-founded society with three types of agents: individuals, insurers, and a government. Nascent to economics literature, we use Reinforcement Learning (RL), closely related to multi-armed bandit problems, to learn the welfare impact of a set of proposed policy interventions per $1 spent on them. The paper rigorously discusses the desirability of the proposed interventions by comparing them against each other on a case-by-case basis. The paper provides a framework for algorithmic policy evaluation using calibrated theoretical models which ca…
University of Virginia（弗吉尼亚大学）；Shanghai Jiao Tong University（上海交通大学）；Microsoft Research（微软研究院）
We study the problem of online influence maximization in social networks. In this problem, a learner aims to identify the set of "best influencers" in a network by interacting with it, i.e., repeatedly selecting seed nodes and observing activation feedback in the network. We capitalize on an important property of the influence maximization problem named network assortativity, which is ignored by most existing works in online influence maximization. To realize network assortativity, we factorize the activation probability on the edges into latent factors on the corresponding nodes, including in…
University of California, Santa Barbara（加州大学圣巴巴拉分校）
We study decentralized stochastic linear bandits, where a network of $N$ agents acts cooperatively to efficiently solve a linear bandit-optimization problem over a $d$-dimensional space. For this problem, we propose DLUCB: a fully decentralized algorithm that minimizes the cumulative regret over the entire network. At each round of the algorithm each agent chooses its actions following an upper confidence bound (UCB) strategy and agents share information with their immediate neighbors through a carefully designed consensus procedure that repeats over cycles. Our analysis adjusts the duration o…
A sequential decision process in which an adaptive radar system repeatedly interacts with a finite-state target channel is studied. The radar is capable of passively sensing the spectrum at regular intervals, which provides side information for the waveform selection process. The radar transmitter uses the sequence of spectrum observations as well as feedback from a collocated receiver to select waveforms which accurately estimate target parameters. It is shown that the waveform selection problem can be effectively addressed using a linear contextual bandit formulation in a manner that is both…
Department of CSE, POSTECH, Pohang, Republic of Korea（POSTECH 电子工程系）；Graduate School of AI, POSTECH, Pohang, Republic of Korea（POSTECH 人工智能研究生院）；Microsoft Research, Beijing, China（微软研究院）；Qiuzhen College, Tsinghua University, Beijing, China（清华大学求真学院）
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；University of Minnesota - Twin Cities（明尼苏达大学双城分校）
This paper studies the adversarial graphical contextual bandits, a variant of adversarial multi-armed bandits that leverage two categories of the most common side information: \emph{contexts} and \emph{side observations}. In this setting, a learning agent repeatedly chooses from a set of $K$ actions after being presented with a $d$-dimensional context vector. The agent not only incurs and observes the loss of the chosen action, but also observes the losses of its neighboring actions in the observation structures, which are encoded as a series of feedback graphs. This setting models a variety o…
