Improving Convergence for Nonconvex Composite Programming

High-dimensional nonconvex problems are popular in today's machine learning and statistical genetics research. Recently, Ghadimi and Lan [1] proposed an algorithm to optimize nonconvex high-dimensional problems. There are several parameters in their algorithm that are to be set before running the algorithm. It is not trivial how to choose these parameters nor there is, to the best of our knowledge, an explicit rule on how to select the parameters to make the algorithm converges faster. We analyze Ghadimi and Lan's algorithm to gain an interpretation based on the inequality constraints for convergence and the upper bound for the norm of the gradient analogue. Our interpretation of their algorithm suggests this to be a damped Nesterov's acceleration scheme. Based on this, we propose an approach on how to select the parameters to improve convergence of the algorithm. Our numerical studies using high-dimensional nonconvex sparse learning problems, motivated by image denoising and statistical genetics applications, show that convergence can be made, on average, considerably faster than that of the conventional ISTA algorithm for such optimization problems with over $10000$ variables should the parameters be chosen using our proposed approach.