We introduce a contractive abstract dynamic programming framework and related policy iteration algorithms, specifically designed for sequential zero-sum games and minimax problems with a general structure. Aside from greater generality, the advantage of our algorithms over alternatives is that they resolve some long-standing convergence difficulties of the ``natural" policy iteration algorithm, which have been known since the Pollatschek and Avi-Itzhak method [PoA69] for finite-state Markov games. Mathematically, this ``natural" algorithm is a form of Newton's method for solving Bellman's equation, but Newton's method, contrary to the case of single-player DP problems, is not globally convergent in the case of a minimax problem, because the Bellman operator may have components that are neither convex nor concave. Our algorithms address this difficulty by introducing alternating player choices, and by using a policy-dependent mapping with a uniform sup-norm contraction property, similar to earlier works by Bertsekas and Yu [BeY10], [BeY12], [YuB13]. Moreover, our algorithms allow a convergent and highly parallelizable implementation, which is based on state space partitioning, and distributed asynchronous policy evaluation and policy improvement operations within each set of the partition. Our framework is also suitable for the use of reinforcement learning methods based on aggregation, which may be useful for large-scale problem instances.
翻译:我们引入了契约式的抽象动态编程框架和相关的政策复制算法, 专门为连续零和游戏和一般结构的小问题设计了这种“ 自然” 政策迭代算法。 除了更普遍外, 我们的算法优于替代方法的优势在于它们解决了“自然” 政策迭代算法的长期趋同困难,自Pollatschek 和 Avi-Itzhak 方法[PoA69] 以来就已知道这些困难。 从数学角度讲, 这种“自然”算法是牛顿解决贝尔曼方程式的方法的一种形式, 但是与单一玩家DP问题的情况相反, 牛顿的方法并不是全球趋同的, 因为贝尔茨克和尤[BeY10] 早期的工作类似, [Y12] 与单一玩家DP问题不同的是, 在小型算法问题上, Bellman 操作可能具有既不相交替又不相交替的成分。 我们的算法解决了这个难题, 与一个统一的 标准性上, Bertsekas and Yu [Y12] 方法, [Yevy train lading cal ligalal develain sal deal develop sal be sal be slading squlation pass be squlation squal be squtefulations