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

We consider stochastic algorithms derived from methods for solving deterministic optimization problems, especially comparison-based algorithms derived from stochastic approximation algorithms with a constant step-size. We develop a methodology for proving geometric convergence of the parameter sequence {θ_n}_n⩾0 of such algorithms. We employ the ordinary differential equation (ODE) method, which relates a stochastic algorithm to its mean ODE, along with a Lyapunov-like function Ψ such that the geometric convergence of Ψ(θ_n) implies – in the case of an optimization algorithm – the geometric convergence of the expected distance between the optimum and the search point generated by the algorithm. We provide two sufficient conditions for Ψ(θ_n) to decrease at a geometric rate: Ψ should decrease “exponentially” along the solution to the mean ODE, and the deviation between the stochastic algorithm and the ODE solution (measured by Ψ) should be bounded by Ψ(θ_n) times a constant. We also provide practical conditions under which the two sufficient conditions may be verified easily without knowing the solution of the mean ODE. Our results are any-time bounds on Ψ(θ_n), so we can deduce not only the asymptotic upper bound on the convergence rate, but also the first hitting time of the algorithm. The main results are applied to a comparison-based stochastic algorithm with a constant step-size for optimization on continuous domains.