A stochastic gradient type algorithm for closed-loop problems

We focus on the numerical solution of closed-loop stochastic problems, and propose a perturbed gradient algorithm to achieve this goal. The main hurdle in such problems is the fact that the control variables are infinite-dimensional, due to, e.g., the information constraints. Alternatively said, control variables are feedbacks, i.e., functions. Such controls have hence to be represented in a finite way in order to solve the problem numerically. In the same way, the gradient of the criterion is itself an infinite-dimensional object. Our algorithm replaces this exact (and unknown) gradient by a perturbed one, which consists of the product of the true gradient evaluated at a random point and a kernel function which extends this gradient to the neighbourhood of the random point. Proceeding this way, we explore the whole space iteration after iteration through random points. Since each kernel function is perfectly known by a small number of parameters, say N, the control at iteration k is perfectly known as an infinite-dimensional object by at most N × k parameters. The main strength of this method is that it avoids any discretization of the underlying space, provided that we can sample as many points as needed in this space. Moreover, our algorithm can take into account the possible measurability constraints of the problem in a new way. Finally, the randomized strategy implemented by the algorithm causes the most probable parts of the space to be the most explored ones, which is a priori an interesting feature. In this paper, we first prove two convergence results of this algorithm in the strongly convex and convex cases, and then give some numerical examples showing the interest of this method for practical stochastic optimization problems.