Proximal Flow Inspired Multi-Step Methods

We investigate a family of approximate multi-step proximal point methods, framed as implicit linear discretizations of gradient flow. The resulting methods are multi-step proximal point methods, with similar computational cost in each update as the proximal point method. We explore several optimization methods where applying an approximate multistep proximal points method results in improved convergence behavior. We also include convergence analysis for the proposed method in several problem settings: quadratic problems, general problems that are strongly or weakly (non)convex, and accelerated results for alternating projections.