混沌、极端主义与乐观主义：游戏学习的量分析
Chaos, Extremism and Optimism: Volume Analysis of Learning in Games
摘要：
我们目前的体积分析乘法权重更新（MWU）和乐观乘法权重更新（OMWU）在零和以及协调游戏。此类分析为这些游戏动力系统提供了新的见解，这似乎很难通过计算机科学和机器学习中的经典技术实现。第一步是不是在原来的空间（行动的单纯形），而是在一个对偶空间（行动的总回报空间）检查这些动态。第二步是探索一组初始条件的体积在根据算法向前推进时如何随时间演变。这让人想起进化博弈论中的方法，其中复制动力学，MWU的连续时间模拟，被称为在所有游戏中始终保持体积。有趣的是，当我们研究离散时间动态时，博弈的选择和算法的选择都起着关键作用。因此，而MWU扩大体积在零和游戏，因此是李雅普诺夫混沌，我们表明，OMWU合同体积，提供了一个替代的理解其已知的收敛行为。然而，我们也证明了一个没有免费午餐类型的定理，在这个意义上说，当检查协调游戏的角色是颠倒的：OMWU膨胀体积指数快速，而MWU合同。使用这些工具，我们证明了两个新的，而负面的性质MWU在零和游戏：（1）极端主义：即使在唯一的完全混合纳什均衡的游戏，系统经常陷入附近的纯策略配置文件，尽管他们显然是不稳定的从博弈论的角度来看。(2)不可避免性：给定任何一组好的点（用你自己对“好”的解释），系统不能无限期地避免坏的点。
英文摘要：
We present volume analyses of Multiplicative Weights Updates (MWU) and Optimistic Multiplicative Weights Updates (OMWU) in zero-sum as well as coordination games. Such analyses provide new insights into these game dynamical systems, which seem hard to achieve via the classical techniques within Computer Science and Machine Learning. The first step is to examine these dynamics not in their original space (simplex of actions) but in a dual space (aggregate payoff space of actions). The second step is to explore how the volume of a set of initial conditions evolves over time when it is pushed forward according to the algorithm. This is reminiscent of approaches in Evolutionary Game Theory where replicator dynamics, the continuous-time analogue of MWU, is known to always preserve volume in all games. Interestingly, when we examine discrete-time dynamics, both the choice of the game and the choice of the algorithm play a critical role. So whereas MWU expands volume in zero-sum games and is thus Lyapunov chaotic, we show that OMWU contracts volume, providing an alternative understanding for its known convergent behavior. However, we also prove a no-free-lunch type of theorem, in the sense that when examining coordination games the roles are reversed: OMWU expands volume exponentially fast, whereas MWU contracts. Using these tools, we prove two novel, rather negative properties of MWU in zero-sum games: (1) Extremism: even in games with unique fully mixed Nash equilibrium, the system recurrently gets stuck near pure-strategy profiles, despite them being clearly unstable from game theoretic perspective. (2) Unavoidability: given any set of good points (with your own interpretation of "good"), the system cannot avoid bad points indefinitely.
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