We combine momentum from machine learning with evolutionary dynamics, where momentum can be viewed as a simple mechanism of intergenerational memory. Using information divergences as Lyapunov functions, we show that momentum accelerates the convergence of evolutionary dynamics including the replicator equation and Euclidean gradient descent on populations. When evolutionarily stable states are present, these methods prove convergence for small learning rates or small momentum, and yield an analytic determination of the relative decrease in time to converge that agrees well with computations. The main results apply even when the evolutionary dynamic is not a gradient flow. We also show that momentum can alter the convergence properties of these dynamics, for example by breaking the cycling associated to the rock-paper-scissors landscape, leading to either convergence to the ordinarily non-absorbing equilibrium, or divergence, depending on the value and mechanism of momentum.

中文翻译：

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动量加速进化动力学

我们将机器学习的动量与进化动力学相结合，其中动量可以被视为一种简单的代际记忆机制。使用信息发散作为李雅普诺夫函数，我们表明动量加速了进化动力学的收敛，包括复制方程和种群的欧几里德梯度下降。当存在进化稳定状态时，这些方法证明了小学习率或小动量的收敛性，并产生了收敛时间的相对减少的分析确定，这与计算非常吻合。即使进化动力学不是梯度流，主要结果也适用。我们还表明，动量可以改变这些动力学的收敛特性，例如通过打破与石头剪刀布景观相关的循环，