Large Deviations for a Class of Nonhomogeneous Markov Chains

Large deviation results are given for a class of perturbed non-homogeneous Markov chains on finite state space which formally includes some stochastic optimization algorithms. Specifically, let {Pn} be a sequence of transition matrices on a finite state space which converge to a limit transition matrix P. Let {Xn} be the associated non-homogeneous Markov chain where Pn controls movement from time n − 1 to n. The main statements are a large deviation principle and bounds for additive functionals of the nonhomogeneous process under some regularity conditions. In particular, when P is reducible, three regimes that depend on the decay of certain " connection " Pn probabilities are identified. Roughly, if the decay is too slow, too fast or in an intermediate range, the large deviation behavior is trivial, the same as the time-homogeneous chain run with P or nontrivial and involving the decay rates. Examples of anomalous behaviors are also given when the approach Pn → P is irregular. Results in the intermediate regime apply to geometrically fast running optimizations, and to some issues in glassy physics. 1. Introduction. The purpose of this paper is to provide some large deviation bounds and principles for a class of nonhomogeneous Markov chains related to some popular stochastic optimization algorithms such as Metropolis and simulated annealing schemes. In a broad sense, these algorithms are stochastic perturbations of steepest descent or " greedy " procedures to find the global minimum of a function H and are in the form of nonhomogeneous Markov chains whose connecting transition kernels converge to a limit kernel associated with steepest descent.