Convex-Concave Backtracking for Inertial Bregman Proximal Gradient Algorithms in Non-Convex Optimization
This work addresses optimization efficiency for researchers and practitioners in non-convex problems, though it is incremental as it builds on existing inertial and backtracking methods.
The paper tackles the challenge of adaptively choosing step size and extrapolation parameters in inertial proximal gradient algorithms for non-convex optimization by introducing a convex-concave backtracking procedure, proving global convergence to critical points and demonstrating improved performance in image processing and machine learning tasks.
Backtracking line-search is an old yet powerful strategy for finding a better step sizes to be used in proximal gradient algorithms. The main principle is to locally find a simple convex upper bound of the objective function, which in turn controls the step size that is used. In case of inertial proximal gradient algorithms, the situation becomes much more difficult and usually leads to very restrictive rules on the extrapolation parameter. In this paper, we show that the extrapolation parameter can be controlled by locally finding also a simple concave lower bound of the objective function. This gives rise to a double convex-concave backtracking procedure which allows for an adaptive choice of both the step size and extrapolation parameters. We apply this procedure to the class of inertial Bregman proximal gradient methods, and prove that any sequence generated by these algorithms converges globally to a critical point of the function at hand. Numerical experiments on a number of challenging non-convex problems in image processing and machine learning were conducted and show the power of combining inertial step and double backtracking strategy in achieving improved performances.