6 research outputs found
A stochastic approximation algorithm with multiplicative step size modification
An algorithm of searching a zero of an unknown function \vphi : \,
\R \to \R is considered: ,\,
, where is the
value of \vphi measured at and is the
measurement error. The step sizes \gam_t > 0 are modified in the
course of the algorithm according to the rule: \, \gamma_t =
\min\{u\, \gamma_{t-1},\, \mstep\} if , and , otherwise, where . That is, at each iteration \gam_t is
multiplied either by or by , provided that the resulting
value does not exceed the predetermined value \mstep. The function
\vphi may have one or several zeros; the random values are
independent and identically distributed, with zero mean and finite
variance. Under some additional assumptions on \vphi, , and
\mstep, the conditions on and guaranteeing a.s.
convergence of the sequence , as well as a.s. divergence,
are determined. In particular, if and for any , one has
convergence for . Due to the
multiplicative updating rule for \gam_t, the sequence
converges rapidly: like a geometric progression (if convergence
takes place), but the limit value may not coincide with, but
instead, approximates one of the zeros of \vphi. By adjusting the
parameters and , one can reach arbitrarily high precision of
the approximation; higher precision is obtained at the expense of
lower convergence rate