A Nonlinear Time-Series Theory for Constant-step SGD with Momentum

Constant-step stochastic gradient descent with momentum (SGDM) augments the parameter recursion with a momentum state that carries geometrically decaying memory of past stochastic gradients. We study this coupled recursion as a nonlinear time series and develop a framework for its stationary behavior and distributional limits. The joint parameter-momentum process forgets its initialization geometrically under a finite $p$th moment for $p>1$, yielding a unique stationary causal solution. Under a second-moment condition, this contraction leads to an invariance principle and quenched Gaussian limit for averaged SGDM, while a small-learning-rate analysis gives a Gaussian limit for the last iterate. In a stochastic-curvature quadratic model, we further study how the rate of initialization forgetting relates to stationary variance and averaged fluctuations.