Most of the literature on learning in games has focused on the restrictive setting where the underlying repeated game does not change over time. Much less is known about the convergence of no-regret learning algorithms in dynamic multiagent settings. In this paper, we characterize the convergence of optimistic gradient descent (OGD) in time-varying games. Our framework yields sharp convergence bounds for the equilibrium gap of OGD in zero-sum games parameterized on natural variation measures of the sequence of games, subsuming known results for static games. Furthermore, we establish improved second-order variation bounds under strong convexity-concavity, as long as each game is repeated multiple times. Our results also apply to time-varying general-sum multi-player games via a bilinear formulation of correlated equilibria, which has novel implications for meta-learning and for obtaining refined variation-dependent regret bounds, addressing questions left open in prior papers. Finally, we leverage our framework to also provide new insights on dynamic regret guarantees in static games.

翻译：大多数关于博弈中学习的文献都局限于基础重复博弈随时间不变的约束性设定。关于动态多智能体环境中无悔学习算法的收敛性，目前所知甚少。本文刻画了乐观梯度下降法在时变博弈中的收敛行为。我们的框架针对零和博弈中参数化的自然变化度量序列，给出了乐观梯度下降法均衡间隙的严格收敛界，并涵盖了静态博弈的已知结论。此外，当每个博弈被重复多次时，我们建立了强凸-强凹条件下的改进二阶变分界。通过双线性关联均衡公式，我们的结果同样适用于时变一般和多人博弈，这对元学习以及获得精细化变分依赖的遗憾界具有全新意义，回答了先前文献中遗留的问题。最后，我们利用该框架为静态博弈中的动态遗憾保证提供了新见解。