Parameter Estimation

The classic machine learning problem involves determining a nonlinear mapping

$${\bf y}_k = G({\bf x}_k,{\bf w})\vspace{-.07in}$$ (3)

where ${\bf x}_k$ is the input, ${\bf y}_k$ is the output, and the nonlinear map $G$ is parameterized by the vector ${\bf w}$. The nonlinear map, for example, may be a feedforward or recurrent neural network (${\bf w}$ are the weights), with numerous applications in regression, classification, and dynamic modeling. Learning corresponds to estimating the parameters ${\bf w}$. Typically, a training set is provided with sample pairs consisting of known input and desired outputs, $\{{\bf x}_k, {\bf d}_k \}$. The error of the machine is defined as ${\bf e}_k = {\bf d}_k - G({\bf x}_k,{\bf w})$, and the goal of learning involves solving for the parameters ${\bf w}$ in order to minimize the expected squared error.

While a number of optimization approaches exist ( e.g., gradient descent using backpropagation), the EKF may be used to estimate the parameters by writing a new state-space representation

$${\bf w}_k = {\bf w}_{k-1} + {\bf u}_k$$ (4)

$${\bf y}_k = G( {\bf x}_{k}, {\bf w}_{k}) + e_k.$$ (5)

where the parameters ${\bf w}_k$ correspond to a stationary process with identity state transition matrix, driven by process noise ${\bf u}_k$ (the choice of variance determines tracking performance). The output ${\bf y}_k$ corresponds to a nonlinear observation on ${\bf w}_k$. The EKF can then be applied directly as an efficient ``second-order'' technique for learning the parameters. In the linear case, the relationship between the Kalman Filter (KF) and Recursive Least Squares (RLS) is given in [3]. The use of the EKF for training neural networks has been developed by Singhal and Wu [4] and Puskorious and Feldkamp [5].