An ODE method to prove the geometric convergence of adaptive stochastic algorithms

We develop a methodology to prove geometric convergence of the parameter sequence $\{\theta_n\}_{n\geq 0}$ of a stochastic algorithm. The convergence is measured via a function $\Psi$ that is similar to a Lyapunov function. Important algorithms that motivate the introduction of this methodology are stochastic algorithms deriving from optimization methods solving deterministic optimization problems. Among them, we are especially interested in analyzing comparison-based algorithms that typically derive from stochastic approximation algorithms with a constant step-size. We employ the so-called ODE method that relates a stochastic algorithm to its mean ODE, along with the Lyapunov-like function $\Psi$ such that the geometric convergence of $\Psi(\theta_n)$ implies---in the case of a stochastic optimization algorithm---the geometric convergence of the expected distance between the optimum of the optimization problem and the search point generated by the algorithm. We provide two sufficient conditions such that $\Psi(\theta_n)$ decreases at a geometric rate. First, $\Psi$ should decrease "exponentially" along the solution to the mean ODE. Second, the deviation between the stochastic algorithm and the ODE solution (measured with the function $\Psi$) should be bounded by $\Psi(\theta_n)$ times a constant. We provide in addition practical conditions that allow to verify easily the two sufficient conditions without knowing in particular the solution of the mean ODE. Our results are any-time bounds on $\Psi(\theta_n)$, so we can deduce not only asymptotic upper bound on the convergence rate, but also the first hitting time of the algorithm. The main results are applied to two comparison-based stochastic algorithms with a constant step-size for optimization on discrete and continuous domains.