Efficient derivative-free Kalman filters for online learning

The extended Kalman filter (EKF) is considered one of the most effective methods for both nonlinear state estimation and parameter estimation (e.g., learning the weights of a neural network). Recently, a number of derivative free alternatives to the EKF for state estimation have been proposed. These include the Unscented Kalman Filter (UKF) [1, 2], the Central Difference Filter (CDF) [3] and the closely related Divided Difference Filter (DDF) [4]. These filters consistently outperform the EKF for state estimation, at an equal computational complexity of O(L3). Extension of the UKF to parameter estimation was presented by Wan and van der Merwe in [5, 6]. In this paper, we further develop these techniques for parameter estimation and neural network training. The extension of the CDF and DDF filters to parameter estimation, and their relation to UKF parameter estimation is presented. Most significantly, this paper introduces efficient square-root forms of the different filters. This enables an O(L2) implementation for parameter estimation (equivalent to the EKF), and has the added benefit of improved numerical stability and guaranteed positive semi-definiteness of the Kalman filter covariances.