  As gradient-free stochastic optimization gains emerging attention for a wide
range of applications recently, the demand for uncertainty quantification of
parameters obtained from such approaches arises. In this paper, we investigate
the problem of statistical inference for model parameters based on
gradient-free stochastic optimization methods that use only function values
rather than gradients. We first present central limit theorem results for
Polyak-Ruppert-averaging type gradient-free estimators. The asymptotic
distribution reflects the trade-off between the rate of convergence and
function query complexity. We next construct valid confidence intervals for
model parameters through the estimation of the covariance matrix in a fully
online fashion. We further give a general gradient-free framework for
covariance estimation and analyze the role of function query complexity in the
convergence rate of the covariance estimator. This provides a one-pass
computationally efficient procedure for simultaneously obtaining an estimator
of model parameters and conducting statistical inference. Finally, we provide
numerical experiments to verify our theoretical results and illustrate some
extensions of our method for various machine learning and deep learning
applications.
