A Perturbed Gradient Algorithm in Hilbert Spaces
Résumé
We propose a perturbed gradient algorithm with stochastic noises to solve a general class of optimization problems. We provide a convergence proof for this algorithm, under classical assumptions on the descent direction, and new assumptions on the stochastic noises. Instead of requiring the stochastic noises to correspond to martingale increments, we only require these noises to be asymptotically so. Furthermore, the variance of these noises is allowed to grow infinitely under the control of a decreasing sequence linked with the gradient stepsizes. We then compare this new approach and assumptions with classical ones in the stochastic approximation literature. As an application of this general setting, we show how the algorithm to solve infinite dimensional stochastic optimization problems recently developped by the authors in another paper is a special case of the following perturbed gradient with stochastic noises.