  This work considers stochastic optimization problems in which the objective
function values can only be computed by a blackbox corrupted by some random
noise following an unknown distribution. The proposed method is based on
sequential stochastic optimization (SSO): the original problem is decomposed
into a sequence of subproblems. Each of these subproblems is solved using a
zeroth order version of a sign stochastic gradient descent with momentum
algorithm (ZO-Signum) and with an increasingly fine precision. This
decomposition allows a good exploration of the space while maintaining the
efficiency of the algorithm once it gets close to the solution. Under Lipschitz
continuity assumption on the blackbox, a convergence rate in expectation is
derived for the ZO-signum algorithm. Moreover, if the blackbox is smooth and
convex or locally convex around its minima, a convergence rate to an
$\epsilon$-optimal point of the problem may be obtained for the SSO algorithm.
Numerical experiments are conducted to compare the SSO algorithm with other
state-of-the-art algorithms and to demonstrate its competitiveness.

Optimization and Control