  We obtain an expansion of the implicit weak discretization error for the
target of stochastic approximation algorithms introduced and studied in
[Frikha2013]. This allows us to extend and develop the Richardson-Romberg
extrapolation method for Monte Carlo linear estimator (introduced in [Talay &
Tubaro 1990] and deeply studied in [Pag{\`e}s 2007]) to the framework of
stochastic optimization by means of stochastic approximation algorithm. We
notably apply the method to the estimation of the quantile of diffusion
processes. Numerical results confirm the theoretical analysis and show a
significant reduction in the initial computational cost.
