Linear Convergence of Proximal Gradient Algorithm with Extrapolation for a Class of Nonconvex Nonsmooth Minimization Problems
This work provides theoretical guarantees for optimization algorithms in nonconvex nonsmooth settings, which is incremental but important for researchers in optimization and machine learning.
The paper tackles the problem of minimizing the sum of a Lipschitz differentiable function and a proper closed convex function, showing that the proximal gradient algorithm with extrapolation converges R-linearly to a stationary point under certain conditions, with the threshold for extrapolation coefficients reducing to 1 for convex cases, leading to R-linear convergence for FISTA with fixed restart.
In this paper, we study the proximal gradient algorithm with extrapolation for minimizing the sum of a Lipschitz differentiable function and a proper closed convex function. Under the error bound condition used in [19] for analyzing the convergence of the proximal gradient algorithm, we show that there exists a threshold such that if the extrapolation coefficients are chosen below this threshold, then the sequence generated converges $R$-linearly to a stationary point of the problem. Moreover, the corresponding sequence of objective values is also $R$-linearly convergent. In addition, the threshold reduces to $1$ for convex problems and, as a consequence, we obtain the $R$-linear convergence of the sequence generated by FISTA with fixed restart. Finally, we present some numerical experiments to illustrate our results.