Online Statistical Inference for Gradient-free Stochastic Optimization

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.