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 gradi- ent 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.