Local Linear Convergence of ISTA and FISTA on the LASSO Problem

We establish local linear convergence bounds for the ISTA and FISTA iterations on the model LASSO problem. We show that FISTA can be viewed as an accelerated ISTA process. Using a spectral analysis, we show that, when close enough to the solution, both iterations converge linearly, but FISTA slows down compared to ISTA, making it advantageous to switch to ISTA toward the end of the iteration processs. We illustrate the results with some synthetic numerical examples.