  In this work, we propose a novel adaptive stochastic gradient-free (ASGF)
approach for solving high-dimensional nonconvex optimization problems based on
function evaluations. We employ a directional Gaussian smoothing of the target
function that generates a surrogate of the gradient and assists in avoiding bad
local optima by utilizing nonlocal information of the loss landscape. Applying
a deterministic quadrature scheme results in a massively scalable technique
that is sample-efficient and achieves spectral accuracy. At each step we
randomly generate the search directions while primarily following the surrogate
of the smoothed gradient. This enables exploitation of the gradient direction
while maintaining sufficient space exploration, and accelerates convergence
towards the global extrema. In addition, we make use of a local approximation
of the Lipschitz constant in order to adaptively adjust the values of all
hyperparameters, thus removing the careful fine-tuning of current algorithms
that is often necessary to be successful when applied to a large class of
learning tasks. As such, the ASGF strategy offers significant improvements when
solving high-dimensional nonconvex optimization problems when compared to other
gradient-free methods (including the so called "evolutionary strategies") as
well as iterative approaches that rely on the gradient information of the
objective function. We illustrate the improved performance of this method by
providing several comparative numerical studies on benchmark global
optimization problems and reinforcement learning tasks.
