We study the performance of optimistic regret-minimization algorithms for both minimizing regret in, and computing Nash equilibria of, zero-sum extensive-form games. In order to apply these algorithms to extensive-form games, a distance-generating function is needed. We study the use of the dilated entropy and dilated Euclidean distance functions. For the dilated Euclidean distance function we prove the first explicit bounds on the strong-convexity parameter for general treeplexes. Furthermore, we show that the use of dilated distance-generating functions allow us to decompose the mirror descent algorithm, and its optimistic variant, into local mirror descent algorithms at each information set. This decomposition mirrors the structure of the counterfactual regret minimization framework. We experimentally compare our algorithms to the popular algorithm CFR+. CFR+ is known to often converge at a rate of $T^{-1}$, or better, in practice. We give an example matrix game where CFR+ experimentally converges at a relatively slow rate of $T^{-0.74}$, whereas optimistic methods converge faster than $T^{-1}$. We go on to show that this fast rate also holds in the Kuhn poker game, which is an extensive-form game. For games with deeper game trees however, we find that CFR+ is still faster. Finally we show that when the goal is minimizing regret, rather than computing a Nash equilibrium, optimistic methods can outperform CFR+, even in deep game trees.

研究了在广泛形式博弈中最小化后悔和计算纳什均衡的乐观后悔最小化算法的性能，研究了扩展形式游戏距离生成函数的使用，证明了扩展欧几里德距离函数具有广义树片段的强凸性参数的第一个显式边界，提出了一种乐观算法可以优化计算效率，这在最小化后悔而不是计算纳什均衡时表现出很好的结果。

扩张式形式博弈中的扩张距离产生函数乐观遗憾最小化