A perturbed gradient algorithm in Hilbert spaces

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 innitely 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 innite dimensional stochastic optimization problems developped in [Barty et al., 2005] is a special case of the following perturbed gradient al-gorithm with stochastic noises. In a second part, we provide analogously a general perturbed gradient algo-rithm to solve saddle point problems, and provide a convergence proof under mild assumptions, in the same spirit as those of the previous theorem