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Extra resources for Advances in Computers, Vol. 24
If one knows the gradient of the function, then one can use the 47 well-known Newton–Raphson method to find the zeros, but in this case one takes the noise-corrupted measurements of the function at different values of . One then makes small corrections to in the estimated direction of zero. The method of stochastic approximation and the theoretical proofs of stability are used in the proofs of convergence for several fundamental algorithms in reinforcement learning. Formulating the problem in a similar form to the previous sections, we get the function and we can write the prediction error and the error in getting to zero as .
The covariance matrix is typically initialized to a relatively large diagonal matrix, and represents the initial uncertainty in the parameter estimate. 38 One can implement the RLS algorithm as in Eq. 20), but the user should be careful that the covariance matrix is always positive definite and symmetric. If the matrix, because of numerical error by repeatedly computing the RLS, ceases to be positive definite and symmetric, then the algorithm will diverge. There are a number of well-developed algorithms to ensure that the matrix remains positive definite.
2 Recursive Least Squares The LS algorithm has been extended to the RLS algorithm. In this case, the parameter estimate is developed as the machine collects the data in real time. In the previous section, all the data was collected first, and then the parameter estimates were computed on the basis of Eq. 6). The RLS algorithm is derived by assuming a solution to the LS algorithm and then adding a single data point. The derivation is shown in Reference . In the RLS implementation, the cost function takes a slightly different form.
Advances in Computers, Vol. 24