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arXiv 2602.16016cs.GTcs.CC
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关于学习纳什均衡的复杂性

On the Complexity of Learning Nash Equilibria

Oliver Biggar, Christos Papadimitriou, Georgios Piliouras

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AI总结：

研究探讨了纳什均衡的学习复杂性，指出即使在非退化游戏中，本地可计算的收敛动态也难以存在，除非复杂性类崩溃。

摘要：

我们知道，除非P=PPAD，否则无法高效计算游戏的纳什均衡。但能否学习它们？是否存在动态（1）可在每个策略配置上高效计算，（2）保证收敛到纳什均衡？这问题与纳什均衡本身一样古老，且早于当前复杂性考虑。最近在MPPS23中证明，一般情况下不存在此类动态；然而，该证明所用的游戏是退化的，并且假设了均匀收敛到连续纳什均衡的强假设。我们指出这两个假设对证明是必要的，因为纳什收敛动态确实存在，它们在非退化游戏中收敛到所有纳什均衡；事实上，我们描述了两种非常不同的动态家族。然而，我们显示这两种家族在本地计算上是不可行的，除非复杂性类崩溃。我们提出了一个复杂性理论的不可能猜想：如果存在本地可计算的纳什收敛动态，则P=PPAD。这是一个结合拓扑和复杂性的新型猜想，因为动态仅需要本地可计算——收敛可能需要指数步骤，只要每一步的工作是多项式。接下来，我们证明任何具有本地可计算的局部可计算李雅普诺夫函数的动态都无法存在，除非PPAD=CLS。最后，对于最一般的不可能猜想，我们提供了一个复杂性理论的解释，说明为何可能难以解决：我们引入了一个证明游戏，以展示黑箱还原无法在多项式时间内区分收敛和非收敛动态。

英文摘要：

We know that the Nash equilibria of a game cannot be computed efficiently unless $P = PPAD$. But can they be learned? Are there dynamics that (1) can be computed efficiently by the players at each strategy profile and (2) are guaranteed to converge to the Nash equilibria? This is a question as ancient as the Nash equilibrium itself, and antedates by many decades current complexity considerations about it. It was recently proved in MPPS23 that no such dynamics can exist in general; however, the game used in that proof is degenerate, and a strong assumption of uniform convergence to a continuum of Nash equilibria is employed. We point out that both assumptions are necessary for that proof, because Nash-convergent dynamics do exist, which converge to all Nash equilibria in non-degenerate games; in fact we describe two very different families of such dynamics. However, we show that both of these families are intractable to compute locally, unless complexity classes collapse. We formulate a complexity theoretic Impossibility Conjecture: if a locally tractable Nash-convergent dynamic exists then $P=PPAD$. This is a novel kind of conjecture, combining topology and complexity, because the dynamic is only required to be tractable locally -- it could take exponentially many steps to converge, so long as the work done at each step is polynomial. Next, we show that any locally tractable dynamic which possesses a locally tractable Lyapunov function cannot exist unless $PPAD = CLS$. Finally, for the most general impossibility conjecture, we provide a complexity-theoretic explanation why it may be difficult to settle: we introduce a Proving Game to demonstrate that black-box reductions cannot distinguish between convergent and non-convergent dynamics in polynomial time.

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