How can the convergence of an LMS filter be accelerated?
Can we do better than the vanilla algorithm?
I'd say there 3 approaches to do so:
Properties of the LMS Filter
There is an optimal step size given you know the spectrum of the correlation matrix. You may have a look at Wikipedia's Least Mean Squares Filter at Convergence and Stability in the Mean.
Some other approaches related to this might be those from Variable Step Size LMS. You may have a look at Review and Comparison of Variable Step Size LMS Algorithms.
Methods of General Optimization
You may use approaches like Line Search and Backtracking for the near optimal per iteration step size.
Methods of Convex Optimization and Stochastic Gradient Descent
You may use optimizers which are used in the Machine Learning / Deep Learning community (Like FISTA) for stochastic gradient descent. Since both the LMS and many algorithms use the Stochastic Gradient Descent they will be valid.