  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 $\{\theta_n\}_{n\geq 0}$ 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 $\Psi$ such that the geometric
convergence of $\Psi(\theta_n)$ 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 $\Psi(\theta_n)$ to decrease at a geometric rate:
$\Psi$ should decrease "exponentially" along the solution to the mean ODE, and
the deviation between the stochastic algorithm and the ODE solution (measured
by $\Psi$) should be bounded by $\Psi(\theta_n)$ 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 $\Psi(\theta_n)$, 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.
