A conceptually appealing approach for learning Extensive-Form Games (EFGs) is to convert them to Normal-Form Games (NFGs). This approach enables us to directly translate state-of-the-art techniques and analyses in NFGs to learning EFGs, but typically suffers from computational intractability due to the exponential blow-up of the game size introduced by the conversion. In this paper, we address this problem in natural and important setups for the \emph{$\Phi$-Hedge} algorithm -- A generic algorithm capable of learning a large class of equilibria for NFGs. We show that $\Phi$-Hedge can be directly used to learn Nash Equilibria (zero-sum settings), Normal-Form Coarse Correlated Equilibria (NFCCE), and Extensive-Form Correlated Equilibria (EFCE) in EFGs. We prove that, in those settings, the \emph{$\Phi$-Hedge} algorithms are equivalent to standard Online Mirror Descent (OMD) algorithms for EFGs with suitable dilated regularizers, and run in polynomial time. This new connection further allows us to design and analyze a new class of OMD algorithms based on modifying its log-partition function. In particular, we design an improved algorithm with balancing techniques that achieves a sharp $\widetilde{\mathcal{O}}(\sqrt{XAT})$ EFCE-regret under bandit-feedback in an EFG with $X$ information sets, $A$ actions, and $T$ episodes. To our best knowledge, this is the first such rate and matches the information-theoretic lower bound.

翻译：学习广泛形式运动会(EFGs) 的概念吸引性方法是将它们转换成普通形式运动会(NFGs) 。 这种方法使我们能够直接翻译NFGs中最先进的技术和分析,学习EFGs,但通常会由于转换带来的游戏规模的飞速打击而出现计算失色。 在本文中,我们解决了 emph{$\Phi$-redge} 算法的自然和重要设置中的这一问题 -- 能够学习大量NFGs类的平衡的通用算法。 我们显示, $Phi$-heet可直接用于学习Nash Equilibria(零和设置)、 普通- Comm Corm Cortal(NFCECE), 以及 EFGserm-Crecontalticlation 和新版本的ODral-lational-ral-lational-lational-lational- disquenal-OD) 。 我们证明, 和新版本的ODral-deal-cal-cal-lationslationslation a brocal-cal-lations brod- drog- drolations a brod a brolations a brolations brolations, 新的 新的 新的 和在新的和在新的和在新版本的自动和新版本的自动和新版本。