Often -- for example in war games, strategy video games, and financial simulations -- the game is given to us only as a black-box simulator in which we can play it. In these settings, since the game may have unknown nature action distributions (from which we can only obtain samples) and/or be too large to expand fully, it can be difficult to compute strategies with guarantees on exploitability. Recent work \cite{Zhang20:Small} resulted in a notion of certificate for extensive-form games that allows exploitability guarantees while not expanding the full game tree. However, that work assumed that the black box could sample or expand arbitrary nodes of the game tree at any time, and that a series of exact game solves (via, for example, linear programming) can be conducted to compute the certificate. Each of those two assumptions severely restricts the practical applicability of that method. In this work, we relax both of the assumptions. We show that high-probability certificates can be obtained with a black box that can do nothing more than play through games, using only a regret minimizer as a subroutine. As a bonus, we obtain an equilibrium-finding algorithm with $\tilde O(1/\sqrt{T})$ convergence rate in the extensive-form game setting that does not rely on a sampling strategy with lower-bounded reach probabilities (which MCCFR assumes). We demonstrate experimentally that, in the black-box setting, our methods are able to provide nontrivial exploitability guarantees while expanding only a small fraction of the game tree.
Certifying Concavity and Monotonicity in Games via Sum-of-Squares Hierarchies
University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Singapore University of Technology and Design（新加坡科技设计大学）；Centre for Quantum Technologies, National University of Singapore（新加坡国立大学量子技术中心）；Archimedes/Athena Research Center（阿尔基米德/雅典研究中心）
Discovering Expert-Level Nash Equilibrium Algorithms with Large Language Models
CFCS, School of Computer Science, Peking University, Beijing, China（计算机科学系，北京大学，北京，中国）；School of Computing and Data Science, The University of Hong Kong, Pokfulam, Hong Kong（计算与数据科学学院，香港大学，薄扶林，香港）
The Graduate School and University Center, The City University of New York（纽约市立大学研究生中心）；University of Birmingham（伯明翰大学）；Brooklyn College, The City University of New York（纽约市立大学布鲁克林学院）
Microsoft Research（微软研究院）；Princeton University（普林斯顿大学）
我们表明，自然类的正则化学习算法与近因偏差的形式实现更快的收敛速度，以近似的效率和粗糙的相关均衡多人范式游戏。当游戏中的每个玩家使用我们班的算法时，他们的个人遗憾以$O(T^{-3/4})$衰减，而效用之和则以$O(T^{-1})$收敛到近似最佳值--这是对最坏情况下$O(T^{-1/2})$率的改进。我们展示了一个黑盒减少类中的任何算法，以实现对对手的$\tilde{O}(T^{-1/2})$率，同时保持对类中的算法更快的速度。我们的结果扩展了[Rakhlin and Shridharan 2013]和[Daskalakis et al. 2014]的结果，他们只分析了特定算法的两人零和游戏。
