LMS is H^∞ optimal

Abstract

We show that the celebrated LMS (Least-Mean Squares) adaptive algorithm is an H^∞ optimal filter. In other words, the LMS algorithm, which has long been regarded as an approximate least-mean squares solution, is in fact a minimizer of the H^∞ error norm and not the H^2 norm. In particular, the LMS minimizes the energy gain from the disturbances to the predicted errors, while the normalized LMS minimizes the energy gain from the disturbances to the filtered errors. Moreover, since these algorithms are central H^∞ filters, they are also risk-sensitive optimal and minimize a certain exponential cost function. We discuss various implications of these results, and show how they provide theoretical justification for the widely observed excellent robustness properties of the LMS filter.

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