  Many data-science problems can be formulated as an inverse problem, where the
parameters are estimated by minimizing a proper loss function. When complicated
black-box models are involved, derivative-free optimization tools are often
needed. The ensemble Kalman filter (EnKF) is a particle-based derivative-free
Bayesian algorithm originally designed for data assimilation. Recently, it has
been applied to inverse problems for computational efficiency. The resulting
algorithm, known as ensemble Kalman inversion (EKI), involves running an
ensemble of particles with EnKF update rules so they can converge to a
minimizer. In this article, we investigate EKI convergence in general nonlinear
settings. To improve convergence speed and stability, we consider applying EKI
with non-constant step-sizes and covariance inflation. We prove that EKI can
hit critical points with finite steps in non-convex settings. We further prove
that EKI converges to the global minimizer polynomially fast if the loss
function is strongly convex. We verify the analysis presented with numerical
experiments on two inverse problems.
