LMS-2: Towards an algorithm that is as cheap as LMS and almost as efficient as RLS

We consider linear prediction problems in a stochastic environment. The least mean square (LMS) algorithm is a well-known, easy to implement and computationally cheap solution to this problem. However, as it is well known, the LMS algorithm, being a stochastic gradient descent rule, may converge slowly. The recursive least squares (RLS) algorithm overcomes this problem, but its computational cost is quadratic in the problem dimension. In this paper we propose a two timescale stochastic approximation algorithm which, as far as its slower timescale is considered, behaves the same way as the RLS algorithm, while it is as cheap as the LMS algorithm. In addition, the algorithm is easy to implement. The algorithm is shown to give estimates that converge to the best possible estimate with probability one. The performance of the algorithm is tested in two examples and it is found that it may indeed offer some performance gain over the LMS algorithm.