  In this paper, we introduce the Ensemble Kalman-Stein Gradient Descent
(EnKSGD) class of algorithms. The EnKSGD class of algorithms builds on the
ensemble Kalman filter (EnKF) line of work, applying techniques from sequential
data assimilation to unconstrained optimization and parameter estimation
problems. The essential idea is to exploit the EnKF as a black box (i.e.
derivative-free, zeroth order) optimization tool if iterated to convergence. In
this paper, we return to the foundations of the EnKF as a sequential data
assimilation technique, including its continuous-time and mean-field limits,
with the goal of developing faster optimization algorithms suited to noisy
black box optimization and inverse problems. The resulting EnKSGD class of
algorithms can be designed to both maintain the desirable property of
affine-invariance, and employ the well-known backtracking line search.
Furthermore, EnKSGD algorithms are designed to not necessitate the subspace
restriction property and variance collapse property of previous iterated EnKF
approaches to optimization, as both these properties can be undesirable in an
optimization context. EnKSGD also generalizes beyond the $L^{2}$ loss, and is
thus applicable to a wider class of problems than the standard EnKF. Numerical
experiments with both linear and nonlinear least squares problems, as well as
maximum likelihood estimation, demonstrate the faster convergence of EnKSGD
relative to alternative EnKF approaches to optimization.
