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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf,len=599
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='11802v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='LG]  27 Jan 2023  Decentralized Online Bandit Optimization on Directed Graphs with Regret  Bounds  Johan ¨Ostman 1 Ather Gattami 1 Daniel Gillblad 1  Abstract  We consider a decentralized multiplayer game,  played over T rounds, with a leader-follower hi-  erarchy described by a directed acyclic graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  For each round, the graph structure dictates the  order of the players and how players observe  the actions of one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' 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 minimiz-  ing the joint pseudo-regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' We present a learn-  ing algorithm inspired by the single-player multi-  armed bandit problem and show that it achieves  sub-linear joint pseudo-regret in the number of  rounds for both adversarial and stochastic ban-  dit rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Furthermore, we quantify the cost  incurred due to the decentralized nature of our  problem compared to the centralized setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Introduction  Decentralized multi-agent online learning concerns agents  that, simultaneously, learn to behave over time in order  to achieve their goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  Compared to the single-agent  setup, novel challenges are present as agents may not  share the same objectives, the environment becomes non-  stationary, and information asymmetry may exist between  agents (Yang & Wang, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  Traditionally, the multi-  agent problem has been addressed by either relying on  a central controller to coordinate the agents’ actions or  to let the agents learn independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  However, access  to a central controller may not be realistic and indepen-  dent learning suffers from convergence issues (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=',  2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' To circumvent these issues, a common approach  is to drop the central coordinator and allow informa-  tion exchange between agents (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  Cesa-Bianchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  Decision-making that involves multiple agents is often  1AI Sweden, Gothenburg, Sweden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Correspondence to: Johan  ¨Ostman <johan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='ostman@ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='se>.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  modeled as a game and studied under the lens of game  theory to describe the learning outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='1  Herein, we  consider games with a leader-follower structure in which  players act consecutively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' For two players, such games  are known as Stackelberg games (Hicks, 1935).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Stackel-  berg games have been used to model diverse learning situ-  ations such as airport security (Balcan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=', 2015), poach-  ing (Sessa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=', 2020), tax planning (Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=', 2020),  and generative adversarial networks (Moghadam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=',  2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  In a Stackelberg game, one is typically con-  cerned with finding the Stackelberg equilibrium, some-  times called Stackelberg-Nash equilibrium, in which the  leader uses a mixed strategy and the follower is best-  responding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' A Stackelberg equilibrium may be obtained by  solving a bi-level optimization problem if the reward func-  tions are known (Sch¨afer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Aussel & Svensson,  2020) or, otherwise, it may be learnt via online learn-  ing techniques (Bai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Zhong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=', 2021), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=',  no-regret algorithms (Shalev-Shwartz, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=',  2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Goktas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  No-regret algorithms have emerged from the single-player  multi-armed bandit problem as a means to alleviate  the exploitation-exploration trade-off (Bubeck & Slivkins,  2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' An algorithm is called no-regret if the difference be-  tween the cumulative rewards of the learnt strategy and the  single best action in hindsight is sublinear in the number  of rounds (Shalev-Shwartz, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' In the multi-armed ban-  dit problem, rewards may be adversarial (based on random-  ness and previous actions), oblivious adversarial (random),  or stochastic (independent and identically distributed) over  time (Auer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=', 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Different assumptions on the ban-  dit rewards yield different algorithms and regret bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  Indeed, algorithms tailored for one kind of rewards are  sub-optimal for others, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=', the EXP3 algorithm due  to Auer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' (2002) yields the optimal scaling for adversar-  ial rewards but not for stochastic rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' For this reason,  best-of-two-worlds algorithms, able to optimally handle  both the stochastic and adversarial rewards, have recently  been pursued and resulted in algorithms with close to op-  timal performance in both settings (Auer & Chiang, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  1The convention is to use agents in learning applications and  players in game theoretic applications, we shall use the game-  theoretic nomenclature in the remainder of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  Decentralized Online Bandit Optimization on Directed Graphs with Regret Bounds  Wei & Luo, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Zimmert & Seldin, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Extensions to  multiplayer multi-armed bandit problems have been pro-  posed in which players attempt to maximize the sum of  rewards by pulling an arm each, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=', (Kalathil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=',  2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Bubeck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  No-regret algorithms are a common element also when an-  alyzing multiplayer games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  For example, in continuous  two-player Stackelberg games, the leader strategy, based  on a no-regret algorithm, converges to the Stackelberg equi-  librium if the follower is best-responding (Goktas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=',  2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' In contrast, if also the follower adopts a no-regret al-  gorithm, the regret dynamics is not guaranteed to converge  to a Stackelberg equilibrium point (Goktas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=', 2022,  Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  In (Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=', 2019), it was shown for two-  player Stackelberg games that a follower playing a, so-  called, mean-based no-regret algorithm, enables the leader  to achieve a reward strictly larger than the reward achieved  at the Stackelberg equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  This result does, how-  ever, not generalize to n-player games as demonstrated  by D’Andrea (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Apart from studying the Stackelberg  equilibrium, several papers have analyzed the regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' For  example, Sessa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' (2020) presented upper-bounds on the  regret of a leader, employing a no-regret algorithm, playing  against an adversarial follower with an unknown response  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Furthermore, Stackelberg games with states were  introduced by Lauffer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' (2022) along with an algorithm  that was shown to achieve no-regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  As the follower in a Stackelberg game observes the leader’s  action, there is information exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' A generalization  to multiple players has been studied in a series of pa-  pers (Cesa-Bianchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' In this line  of work, players with a common action space form an ar-  bitrary graph and are randomly activated in each round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  Active players share information with their neighbors by  broadcasting their observed loss, previously received neigh-  bor losses, and their current strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' The goal of the play-  ers is to minimize the network regret, defined with respect  to the cumulative losses observed by active players over  the rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' The players, however, update their strategies  according to their individually observed loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Although we  consider players connected on a graph, our work differs  significantly from (Cesa-Bianchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' 2021),  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=', we allow only actions to be observed between players  and the players update their strategies based on a common  bandit reward rather than an individual reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  Contributions:  We introduce the joint pseudo-regret,  defined with respect to the cumulative reward where  all the players observe the same bandit-reward in each  round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' We provide an online learning-algorithmfor general  consecutive-play games that relies on no-regret algorithms  developed for the single-player multi-armed bandit prob-  lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' The main novelty of our contribution resides in the  joint analysis of players with coupled rewards where we  derive upper bounds on the joint pseudo-regret and prove  our algorithm to be no-regret in the adversarial setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Fur-  thermore, we quantify the penalty incurred by our decen-  tralized setting in relation to the centralized setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Problem formulation  In this section, we formalize the consecutive structure of  the game and introduce the joint pseudo-regret that will  be used as a performance metric throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' We consider  a decentralized setting where, in each round of the game,  players pick actions consecutively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' The consecutive nature  of the game allows players to observe preceding players’  actions and may be modeled by a DAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' For example, in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' 1, a seven-player game is illustrated in which player 1  initiates the game and her action is observed by players 2, 5,  and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' The observations available to the remaining players  follow analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Note that for a two-player consecutive  game, the DAG models a Stackelberg game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  We let G = (V, E) denote a DAG where V denotes the ver-  tices and E denotes the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' For our setting, V constitutes  the n different players and E = {(j, i) : j → i, j ∈ V, i ∈  V} describes the observation structure where j → i indi-  cates that player i observes the action of player j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Accord-  ingly, a given player i ∈ V observes the actions of its direct  parents, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=', players j ∈ Ei = {k : (k, i) ∈ E}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Further-  more, each player i ∈ V is associated with a discrete action  space Ai of size Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' We denote by πi(t), the mixed strat-  egy of player i over the action space Ai in round t ∈ [T ]  such that πi(t) = a with probability pi,a for a ∈ Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' In  the special case when pi,a = 1 for some a ∈ Ai, the strat-  egy is referred to as pure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Let AB denote the joint action  space of players in a set B given by the Cartesian product  AB = �  i∈B Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' If a player i has no parents, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=', Ei = ∅,  we use the convention |AEi| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  We consider a collaborative setting with bandit rewards  given by a mapping rt : AV → [0, 1] in each round t ∈ [T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  The bandit rewards are assumed to be adversarial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Let C de-  note a set of cliques in the DAG (Koller & Friedman, 2009,  Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='13) and let Nk ∈ C for k ∈ [|C|] denote the players  in the kth clique in C with joint action space ANk such that  Nk ∩ Nj = ∅ for j ̸= k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' For a joint action a(t) ∈ AV,  we consider bandit rewards given by a linear combination  of the clique-rewards as  rt(a(t)) =  |C|  �  k=1  βkrk  t (P k(a(t))),  (1)  where rk  t  :  ANk  →  [0, 1], βk  ≥  0 is the weight  of the kth clique reward such that �|C|  k=1 βk  =  1,  and P k(a(t)) denotes the joint action of the players in  Nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' As an example, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' 2 highlights the cliques C =  Decentralized Online Bandit Optimization on Directed Graphs with Regret Bounds  1  2  3  4  5  6  7  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' A game with seven players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  1  2  3  4  5  6  7  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Colored cliques comprising the bandit reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  {{2, 3, 4}, {1, 5}, {6}, {7}} and we have, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=', N1  =  {2, 3, 4}, and P 1(a(t)) = (a2(t), a3(t), a4(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Note that  each player influences only a single term in the reward (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  In each round t ∈ [T ], the game proceeds as follows for  player i ∈ V:  1) the player is idle until the actions of all parents in Ei  have been observed,  2) the player picks an action ai(t) ∈ Ai according to its  strategy πi(t),  3) once all the n players in V have chosen an action,  the player observes the bandit reward rt(a(t)) and up-  dates its strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  The goal of the game is to find policies {πi(t)}n  i=1 that de-  pend on past actions and rewards in order to minimize the  joint pseudo-regret R(T ) which is defined similarly to the  pseudo regret (Shalev-Shwartz, 2012, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='2) as  R(T ) = r(a⋆) − E  � T  �  t=1  rt(a(t))  �  ,  (2)  where  r(a⋆) = max  a∈AV E  � T  �  t=1  rt(a)  �  ,  and the expectations are taken with respect to the rewards  and the player actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='2 Note that r(a⋆) corresponds to the  largest expected reward obtainable if all players use pure  strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Hence, the pseudo-regret in (2) quantifies the  difference between the expected reward accumulated by the  learnt strategies and the reward-maximizing pure strategies  in hindsight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  Our problem formulation pertains to a plethora of appli-  cations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  Examples include resource allocation in cog-  nitive radio networks where available frequencies are  obtained via channel sensing (Janatian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=', 2015) and  semi-autonomous vehicles with adaptive cruise control,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=', vehicles ahead are observed before an action is de-  cided (Marsden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=', 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  Also recently, the impor-  tance of coupled rewards and partner awareness through  implicit communications, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=', by observation, has been  highlighted in human-robot and human-AI collaborative  settings (Bıyık et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Furthermore, our formulation  is applicable in simple scenarios within to reinforcement  learning (Ibarz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  As will be shown in the next section, any no-regret al-  gorithm can be used as a building block for the games  considered herein to guarantee a sub-linear pseudo-regret  in the number of rounds T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  As our goal is to study  the joint pseudo-regret (2) for adversarial, we start from  a state-of-the-art algorithm for the adversarial multi-  armed bandit problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' In particular, we will utilize the  TSALLIS-INF algorithm that guarantees a pseudo-regret  with the optimal scaling in the adversarial single-player set-  ting (Zimmert & Seldin, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Analysis of the joint pseudo-regret  Our analysis of the joint pseudo-regret builds upon learning  algorithms for the single-player multi-armed bandit prob-  lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' First, let us build intuition on how to use a multi-  armed bandit algorithm in the DAG-based game described  in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Consider a 2-player Stackelberg game where  the players choose actions from A1 and A2, respectively,  and where player 2 observes the actions of player 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' For  simplicity, we let player 1 use a mixed strategy whereas  player 2 is limited to a pure strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Furthermore, con-  sider the rewards to be a priori known by the players and let  T = 1 for which the Stackelberg game may be viewed as a  bi-level optimization problem (Aussel & Svensson, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  In this setting, the action of player 1 imposes a Nash game  on player 2 whom attempts to play optimally given the ob-  servation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Hence, player 2 has A1 pure strategies, one for  each of the A1 actions of player 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  We may generalize this idea to the DAG-based multiplayer  game with unknown bandit-rewards and T ≥ 1 to achieve  2This is called pseudo-regret as r(a⋆) is obtained by a maxi-  mization outside of the expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  Decentralized Online Bandit Optimization on Directed Graphs with Regret Bounds  no-regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Indeed, a player i ∈ V may run |AEi| different  multi-armed bandit algorithms, one for each of the joint  actions of its parents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  Algorithm 1 illustrates this idea  in conjunction with the TSALLIS-INF update rule intro-  duced by Zimmert & Seldin (2021), which is given in Al-  gorithm 2 for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='3 In particular, for the 2-player  Stackelberg game, the leader runs a single multi-armed ban-  dit algorithm whereas the follower runs A1 learning algo-  rithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' For simplicity, Algorithm 1 assumes that player i  knows the size of the joint action space of its parents, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=',  |AEi|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Dropping this assumption is straightforward: simply  keep track of the observed joint actions and initiate a new  multi-armed bandit learner upon a unique observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  Algorithm 1 Learning algorithm of player i ∈ V  1: Input: for ease of notation, let the actions in AEi be  labeled as 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=', |AEi|  2: initialize cumulative loss Lk ← 0 ∈ RAi for k ∈  [|AEi|]  3: initialize fixed-point xk ← 0 for k ∈ [|AEi|]  4: initialize counter nk ← 0 for k ∈ [|AEi|]  5: for t = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' T do  6:  observe the joint action j ∈ [|AEi|] of the preceding  players  7:  increase counter nj ← nj + 1  8:  obtain  new  strategy  and  new  fixed-point  (πi(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' xj) ← TSALLIS-INF(nj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Lj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' xj)  9:  play action ai(t) ∼ πi(t)  10:  observe the joint bandit-reward rt(a(t))  11:  update the cumulative loss for all k ∈ [Ai] as  Lj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='k ← Lj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='k + 1{ai(t) = k}(1 − rt(a(t)))/pk  12: end for  Algorithm 2 Strategy update for player i ∈ V  1: Input: time step t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' cumulative rewards L ∈ RAi  + ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' pre-  vious fixed point x Output strategy πi(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' fixed point  x  2: set learning rate η ← 2  �  1/t  3: repeat  4:  pj ← 4(η(Lj − x))−2 for all j ∈ [Ai]  5:  x ← x −  ��Ai  j=1 pj − 1  �  /  �  η �Ai  j=1 p3/2  j  �  6: until convergence  7: update strategy πi(t) ← (p1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' , pAi)  Next, we go on to analyze the joint pseudo-regret of Algo-  rithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' First, we present a result on the pseudo-regret for  the single-player multi-armed bandit problem that will be  used throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  3The original TSALLIS-INF Algorithm is given in terms of  losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' To use rewards, one may simply use the relationship l =  1 − r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='1 (Pseudo-regret of TSALLIS-INF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Consider  a single-player multi-armed bandit problem with A1 arms,  played over T rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Let the player operate according to  Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Then, the pseudo-regret satisfies  R(T ) ≤ 4  �  A1T + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' For a single player, E1 = ∅ and we have |AE1| =  1 by convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Hence, our setting becomes equivalent  to that of Zimmert & Seldin (2021, Th 1) and the result  follows thereof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  Next, we consider a two-player Stackelberg game with  joint bandit-rewards defined over a two-player clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' We  have the following upper bound on the joint pseudo-regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='2 (Joint pseudo-regret over cliques of size 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  Consider a 2-player Stackelberg game with bandit-rewards,  given by (1), defined over a single clique containing both  players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Furthermore, let each of the players follow Algo-  rithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Then, the joint pseudo-regret satisfies  R(T ) ≤ 4  �  A1A2T + 4  �  A1T + A1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Without loss of generality, let player 2 observe the  actions of player 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Let a1(t) ∈ A1 and a2(t) ∈ A2 denote  the actions of player 1 and player 2, respectively, at time t ∈  [T ] and let a⋆  1 and a⋆  2(a1) denote the reward-maximizing  pure strategies of the players in hindsight, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=',  a⋆  1 = arg max  a1∈A1 E  � T  �  t=1  rt(a1, a⋆  2(a1))  �  ,  (3)  a⋆  2(a1) = arg max  a2∈A2 E  � T  �  t=1  rt(a1, a2)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  (4)  Note that the optimal joint decision in hindsight is given by  (a⋆  1, a⋆  2(a⋆  1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' The joint pseudo-regret is given by  R(T ) =  T  �  t=1  E [rt(a⋆  1, a⋆  2(a⋆  1)) − rt(a⋆  1, a2(t))]  + E [rt(a⋆  1, a2(t)) − rt(a1(t), a2(t))]  ≤  T  �  t=1  max  at∈A1 E [rt(at, a⋆  2(at)) − rt(at, a2(t))]  + E  � T  �  t=1  rt(a⋆  1, a2(t)) − rt(a1(t), a2(t))  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  (5)  Next, let  a+  1 (t) = arg max  at∈A1 E [rt(at, a⋆  2(at)) − rt(at, a2(t))]  and let Ta = {t : a+  1 (t) = a}, for a ∈ A1, denote all the  rounds that player 1 chose action a and introduce Ta = |Ta|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  Decentralized Online Bandit Optimization on Directed Graphs with Regret Bounds  Then, the first term in (5) is upper-bounded as  T  �  t=1  max  at∈A1 E [rt(at, a⋆  2(at)) − rt(at, a2(t))]  =  �  a∈A1  �  t∈Ta  E [rt(a, a⋆  2(a)) − rt(a, a2(t))]  ≤  �  a∈A1  4  �  A2Ta + 1  (6)  ≤  max  �  a Ta=T  �  a∈A1  4  �  A2Ta + 1  = 4  �  A1A2T + A1  (7)  where (6) follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='1 and because player 2  follows Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Note that the actions in Ta may not be  consecutive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' However, as we consider adversarial rewards,  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='1 is still applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  Next, we consider the second term in (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Note that, accord-  ing to (4), a⋆  1 is obtained from the optimal pure strategies  in hindsight of both the players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Let  a◦  1 = arg max  a1∈A1  T  �  t=1  E [rt(a1, a2(t))]  and note that  E  � T  �  t=1  rt(a⋆  1, a2(t))  �  ≤ E  � T  �  t=1  rt(a◦  1, a2(t))  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  By adding and subtracting rt(a◦  1, a2(t)) to the second term  in (5), we get  E  � T  �  t=1  rt(a⋆  1, a2(t)) − rt(a1(t), a2(t))  �  ≤ E  � T  �  t=1  rt(a◦  1, a2(t)) − rt(a1(t), a2(t))  �  ≤ 4  �  A1T + 1  (8)  where the last equality follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' The re-  sult follows from (7) and (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  From Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='2, we note that the joint pseudo-regret  scales with the size of the joint action space as R(T ) =  O(√A1A2T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' This is expected as a centralized version  of the cooperative Stackelberg game may be viewed as  a single-player multi-armed bandit problem with A1A2  arms where, according to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='1, the pseudo-regret  is upper-bounded by 4√A1A2T + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  Hence, from  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='2, we observe a penalty of 4√A1T + A1  due to the decentralized nature of our setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  More-  over, in the single-player setting, Algorithm 2 was shown  in Zimmert & Seldin (2021) to achieve the same scaling as  the lower bound in Cesa-Bianchi & Lugosi (2006, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  Hence, Algorithm 1 achieves the optimal scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Next, we  extend Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='2 to cliques of size larger than two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='3 (Joint pseudo-regret over a clique of arbitrary  size).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Consider a DAG-based game with bandit rewards  given by (1), defined over a single clique containing m  players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Let each of the players operate according to Al-  gorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Then, the joint pseudo-regret satisfies  R(T ) ≤ 4  √  T  m  �  i=1  i�  k=1  �  Ak +  m−1  �  i=1  i�  k=1  Ak + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Let Rub(T, m) denote an upper bound on the joint  pseudo-regret when the bandit-reward is defined over a  clique containing m players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' From Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='1 and Theo-  rem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='2, we have that  Rub(T, 1) = 4  �  A1T + 1  Rub(T, 2) = 4  �  A1T + 4  �  A1A2T + A1 + 1,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Therefore, we form an induction hypothesis  as  Rub(T, m) = 4  √  T  m  �  i=1  i�  k=1  �  Ak +  m−1  �  i=1  i�  k=1  Ak + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' (9)  Assume that (9) is true for a clique containing m − 1  players and add an additional player, assigned player in-  dex 1, whose actions are observable to the original m − 1  players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  The m players now form a clique C of size  m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  Let a(t) ∈ AC denote the joint action of all the  players in the clique at time t ∈ [T ] and let a−i(t) =  (a1(t), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' , ai−1(t), ai+1(t), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' , am(t)) ∈ AC\\i denote  the joint action excluding the action of player i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Further-  more, let  a⋆  1 = arg max  a1∈A1 E  � T  �  t=1  rt(a1, a⋆  −1(a1))  �  a⋆  −1(a1) = arg max  a∈AC\\1 E  � T  �  t=1  rt(a1, a)  �  denote the optimal actions in hindsight of player 1 and the  optimal joint action of the original m − 1 players given the  action of player 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' The optimal joint action in  hindsight is given as a⋆ = (a⋆  1, a⋆  −1(a⋆  1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Following the  Decentralized Online Bandit Optimization on Directed Graphs with Regret Bounds  steps in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='2 verbatim,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' we obtain  R(T ) =  T  �  t=1  E [rt(a⋆) − rt(a⋆  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' a−1(t))]  + E [rt(a⋆  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' a−1(t)) − rt(a(t))]  ≤  T  �  t=1  max  a1 E  �  rt(a1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' a⋆  −1(a1)) − rt(a1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' a−1(t))  �  +  T  �  t=1  E [rt(a⋆  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' a−1(t)) − rt(a1(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' a−1(t))]  ≤  �  a∈A1  �  t∈Ta  E  �  rt(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' a⋆  −1(a)) − rt(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' a−1(t))  �  +  T  �  t=1  E [rt(a◦  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' a−1(t)) − rt(a1(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' a−1(t))]  ≤  �  a∈A1  Rub(Ta,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' m − 1) + 4  �  A1T + 1  ≤ A1Rub(T/A1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' m − 1) + 4  �  A1T + 1  (10)  where Ta,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Ta,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' and a◦  n are defined analogously as in the  proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' By using the induction hypothe-  sis (9) in (10) and by accounting for the original m − 1  players being indexed from 2 to m, we obtain  R(T ) ≤ A1  �  4  �  T/A1  m  �  i=2  i�  k=2  �  Ak +  m−1  �  i=2  i�  k=2  Ak + 1  �  + 4  �  A1T + 1  = Rub(T, m)  which is what we wanted to show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  As in the two-player game,  the joint pseudo-regret  of  Algorithm 1  achieves the  optimal scaling,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=',  R(T )  =  O(  √  T �m  k=1  √Ak), but exhibits a penalty  due to the decentralized setting which is equal to  4  √  T �m−1  i=1  �i  k=1  √Ak + �m−2  i=1  �i  k=1 Ak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  Up until this point, we have considered the pseudo-regret  when the bandit-reward (1) is defined over a single clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  The next theorem leverages the previous results to provide  an upper bound on the joint pseudo-regret when the bandit-  reward is defined over an arbitrary number of independent  cliques in the DAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='4 (Joint pseudo-regret in DAG-based games).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  Consider a DAG-based game with bandit rewards given as  in (1) and let C contain a collection of independent cliques  associated with the DAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Let each player operate accord-  ing to Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Then, the joint pseudo-regret satisfies  R(T ) = O  ��  T max  k∈[|C|] |ANk|  �  where ANk denotes the joint action-space of the players in  the kth clique Nk ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Let Nk ∈ C denote the players belonging to the kth  clique in C with joint action space ANk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' The structure of (1)  allows us to express the joint pseudo-regret as  R(T ) = E  � T  �  t=1  rt(a⋆) − rt(a(t))  �  ≤  |C|  �  k=1  βkE  � T  �  t=1  rk  t (a⋆  k) − rk  t (P k(a(t)))  �  (11)  where  a⋆ = arg max  a∈AV E  � T  �  t=1  rt(a)  �  ,  a⋆  k = arg max  a∈ANk  E  � T  �  t=1  rk  t (a)  �  ,  and the inequality follows since E  ��T  t=1 rk  t (P k(a⋆))  �  ≤  E  ��T  t=1 rk  t (a⋆  k)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Now, for each clique Nk ∈ C, let the  player indices in Nk be ordered according to the order of  player observations within the clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  As Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='3  holds for any Nk ∈ C, we may, with a slight abuse of nota-  tion, bound the joint pseudo-regret of each clique as  R(T ) ≤  |C|  �  k=1  βkRub(T, Nk) ≤ max  k∈[|C|] βkRub (T, Nk)  where Rub(T, Nk) follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='3 as  Rub(T, Nk) = 4  √  T  �  i∈Nk  �  j≤i,j∈Nk  �  Aj  +  �  i∈N −  k  �  j≤i,j∈N −  k  Aj + 1  where N −  k excludes the last element in Nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' The result fol-  lows as Rub(T, Nk) = O(  �  T |ANk|) where the βk has  been consumed in the prefactor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Numerical results  The experimental setup in this section is inspired by the  socio-economic simulation in (Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='4  We  consider a simple taxation game where one player acts as  a socio-economic planner and the remaining M players act  4The source code of our experiments is available on  https://anonymous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='4open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='science/r/bandit_optimization_dag-242C/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  Decentralized Online Bandit Optimization on Directed Graphs with Regret Bounds  as workers that earn an income by performing actions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=',  constructing houses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' The socio-economic planner divides  the possible incomes into N brackets where [βi−1, βi] de-  notes the ith bracket with β0 = 0 and βN = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' In each  round t ∈ [T ], the socio-economic planner picks an action  ap(t) = (ap,1(t), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' , ap,N(t)) that determines the taxation  rate where ap,i(t) ∈ Ri denotes the the marginal taxation  rate in income bracket i and Ri is a finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' We use the  discrete set Ap = �N  i=1 Ri of size Ap to denote the action  space of the planner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  In each round, the workers observe the taxation policy  ap(t) ∈ Ap and choose their actions consecutively, see  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Worker j ∈ [M] takes actions aj(t) ∈ Aj where  Aj is a finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' A chosen action aj(t) ∈ Aj translates  into a tuple (xj(t), ˜lj(t)) consisting of a gross income and  a marginal default labor cost, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Furthermore,  each worker has a skill level sj that serves as a divisor of  the default labor, resulting in an effective marginal labor  lj(t) = ˜lj(t)/sj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Hence, given a common action, high-  skilled workers exhibit less labor than low-skilled workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  The gross income xj(t) of worker j in round t is taxed ac-  cording to ap(t) as  ξ(xj(t)) =  N  �  i=1  ap,i(t)(βi − βi−1)1{xj(t) > βi}  + (xj(t) − βi−1)1{xj(t) ∈ [βi−1, βi]}  where ap,i(t) is the taxation rate of the ith income bracket  and ξ(xj(t)) denotes the collected tax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Hence, worker j’s  cumulative net income zj(t) and cumulative labor ℓj(t) in  round t are given as  zj(t) =  t  �  u=1  xj(u) − ξ(xj(u)),  ℓj(t) =  t  �  u=1  lj(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  In round t, the utility of worker j depends on the cumula-  tive net income and the cumulative labor as  rj  t (zj(t), ℓj(t)) = (zj(t))1−η − 1  1 − η  − ℓj(t)  (12)  where η > 0 determines the non-linear impact of income.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  An example of the utility function in (12) is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' 4  for η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='3, income xj(t) = 10, and a default marginal la-  bor ˜lj(t) = 1 at different skill levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' It can be seen that the  utility initially increases with income until a point at which  the cumulative labor outweighs the benefits of income and  the worker gets burnt out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  We consider bandit-rewards defined with respect to the  Socio-economic planner  1  2  3  workers  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Socio-economic setup with 4 players among which 3 are  designated workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  100  101  102  103  104  0  200  400  600  800  1,000  houses built  worker utility  s = 1  s = 2  s = 3  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Example of utility functions for different skill levels  when xj(t) = 10 and ˜ℓj(t) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  worker utilities and the total collected tax as  rt(ap(t), a1(t), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' , aM(t)) =  1  (M + 1)  \uf8ee  \uf8f0  M  �  j=1  wrj  t (zj(t), ℓj(t)) + wp  M  �  j=1  ξ(xj(t))  \uf8f9  \uf8fb  (13)  where the weights trade off worker utility for the col-  lected tax and satisfy Mw + wp  =  M + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  The  individual rewards are all normalized to [0, 1], hence,  rt(ap(t), a1(t), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' , aM(t)) ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  For the numerical experiment, we consider N = 2 in-  come brackets where the boundaries of the income brackets  are {0, 14, ∞} and the socio-economic planner chooses a  marginal taxation rate from R = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='5} in each  income bracket, hence, Ap = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' We consider M = 3 work-  ers with the same action set A of size 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Consequently,  the joint action space is of size 243.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Furthermore, we let  the skill level of the workers coincide with the worker in-  dex, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=', sj = j for j ∈ [M].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  Simply, workers able  to observe others have higher skill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' The worker actions  translate to a gross marginal income and a marginal la-  bor as aj(t) → (xj(t), lj(t)) where xj(t) = 5aj(t) and  Decentralized Online Bandit Optimization on Directed Graphs with Regret Bounds  lj(t) = aj(t)/sj for aj(t) ∈ {1, 2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Finally, we set  η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='3 and let w = 1/M and wp = M to model a sit-  uation where the collected tax is preferred over workers’  individual utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  The joint pseudo-regret of the socio-economic simulation is  illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' 5 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' 6 (different scales) along with  the upper bound in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' We collect 100 realiza-  tions of the experiment and, along with the pseudo-regret  R(T ), two standard deviations are also presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' It can  be seen that the players initially explore the action space  and are able to eventually converge on an optimal strategy  from a pseudo-regret perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' The upper bound in the  figures is admittedly loose and does not exhibit the same  asymptotic decay as the simulation due to different con-  stants in the scaling law, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' However, it remains  valuable as it provides an asymptotic no-regret guarantee  for the learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  100  101  102  103  104  105  106  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='5  T  Regret  Upper bound  R(T )  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Pseudo regret vs the upper bound in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='4 (linear  scale).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Conclusion  We have studied multiplayer games with joint bandit-  rewards where players execute actions consecutively and  observe the actions of the preceding players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  We intro-  duced the notion of joint pseudo-regret and presented an  algorithm that is guaranteed to achieve no-regret for adver-  sarial bandit rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' A bottleneck of many multi-agent  algorithms is that the complexity scales with the joint ac-  tion space (Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=', 2021) and our algorithm is no ex-  ception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' An interesting venue of further study is to find  algorithms that have more benign scaling properties, see  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=', (Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Daskalakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  Further-  more, recent results on correlated multi-armed bandits have  demonstrated that multi-armed bandits with many arms  may become significantly more feasible if one is able to  101  102  103  104  105  106  10−3  10−2  10−1  100  101  102  T  Regret  Upper bound  R(T )  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Pseudo regret vs the upper bound in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='4 (log-  scale).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  exploit dependencies among arms (Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' It  would be interesting to explore how the scaling of our algo-  rithm is affected by modelling and exploiting dependencies  among players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
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+page_content=', and Jordan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  Can  reinforcement learning find Stackelberg-Nash equilibria  in general-sum Markov games with myopic followers?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=',  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='13521.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content='  Zimmert, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' and Seldin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Tsallis-inf: An optimal algo-  rithm for stochastic and adversarial bandits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}
+page_content=' Journal of  Machine Learning Research, 22(28):1–49, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtFKT4oBgHgl3EQfZS4v/content/2301.11802v1.pdf'}