Adaptive fiiler (LMS) step size bounds
You can analyse the input online, and get some rough estimates on its power, energy, pdf, autocorrelation matrix, etc. and also its eigenvalues of (time-varying) autocorrelation matrix. Then you may estimate the associated upper bound for the step-size of LMS.
Nevertheless, finding the optimal step-size in applications within unknown environments is not easy. So many of the LMS variations are based on different approaches for efficiently, and reliably finding the best step-sizes.