Tracking Time Varying Parameters Via Online Simplified Maximum Likelihood

Usually, log-likelihood functions fail to satisfy the classical assumptions of strong convexity and Lipschitz-continuity of the gradient (as well as many of their mild counterparts) that are common in general convergence results for stochastic gradient descent algorithms. Therefore, the use of gradient descent schemes to track the maxima of a sequence of objective log-likelihood functions suffers from the lack of theoretical results that guarantee the validity of the method. In this paper, we propose a simplified online scheme to track unknown dynamic parameters that are the optima of a sequence of objective log-likelihood functions. Under a Lipschitz assumption on the time varying optimum we demonstrate that our estimator achieves mean square convergence up to a neighborhood of the optimum, and we establish that the Lipschitz continuity assumption is necessary when a specific desirable property is imposed. The method is inspired by a Taylor expansion of the log-likelihood function around the maximum likelihood estimator, and rigorously justified by the expression for the Riemannian gradient of the log-likelihood of a multivariate Gaussian distribution.