The gradient descent is the simplest optimisation algorithm represented by gradient dynamics in a potential. When the input data is finite, gradient descent dynamics fluctuates due to the finite size effects, and is called stochastic gradient descent. In this paper, we study stability of stochastic gradient descent dynamics from the viewpoint of dynamical systems theory. Learning is characterised as nonautonomous dynamics driven by uncertain input from the external, and as multi-scale dynamics which consists of slow memory dynamics and fast system dynamics. When the uncertain input sequences are modelled by stochastic processes, dynamics of learning is described by a random dynamical system. In contrast to the traditional Fokker-Planck approaches [5, 15], the random dynamical system approaches enable the study not only of stationary distributions and global statistics, but also of the pathwise structure of stochastic dynamics. Based on nonautonomous and random dynamical system theory, it is possible to analyse stability and bifurcation in machine learning.

We derive a class of macroscopic differential equations that describe collective adaptation, starting from a discrete-time stochastic microscopic model. The behavior of each agent is a dynamic balance between adaptation that locally achieves the best action and memory loss that leads to randomized behavior. We show that, although individual agents interact with their environment and other agents in a purely self-interested way, macroscopic behavior can be interpreted as game dynamics. Application to several familiar, explicit game interactions shows that the adaptation dynamics exhibits a diversity of collective behaviors. The simplicity of the assumptions underlying the macroscopic equations suggests that these behaviors should be expected broadly in collective adaptation. We also analyze the adaptation dynamics from an information-theoretic viewpoint and discuss self-organization induced by information flux between agents, giving a novel view of collective adaptation.

Noise-induced coherent oscillations in randomly connected neural networks.

Noise-induced coherent oscillations in randomly connected neural networks.