Accelerating Fixed-Point Algorithms in Statistics and Data Science: A State-of-Art Review

Fixed-point algorithms are popular in statistics and data science due to their simplicity, guaranteed convergence, and applicability to high-dimensional problems. Well-known examples include the expectation-maximization (EM) algorithm, majorization-minimization (MM), and gradient-based algorithms like gradient descent (GD) and proximal gradient descent. A characteristic weakness of these algorithms is their slow convergence. We discuss several state-of-art techniques for accelerating their convergence. We demonstrate and evaluate these techniques in terms of their efficiency and robustness in six distinct applications. Among the acceleration schemes, SQUAREM shows robust acceleration with a mean 18-fold speedup. DAAREM and restarted-Nesterov schemes also demonstrate consistently impressive accelerations. Thus, it is possible to accelerate the original fixed-point algorithm by using one of SQUAREM, DAAREM, or restarted-Nesterov acceleration schemes. We describe implementation details and software packages to facilitate the application of the acceleration schemes. We also discuss strategies for selecting a particular acceleration scheme for a given problem.