Optimization of Graph Total Variation via Active-Set-based Combinatorial Reconditioning
This work addresses optimization challenges in machine learning and computer vision, but it is incremental as it builds on existing proximal algorithms with a novel preconditioning approach.
The authors tackled the problem of structured convex optimization on weighted graphs by proposing an adaptive preconditioning strategy for proximal algorithms, which achieves competitive performance in numerical experiments.
Structured convex optimization on weighted graphs finds numerous applications in machine learning and computer vision. In this work, we propose a novel adaptive preconditioning strategy for proximal algorithms on this problem class. Our preconditioner is driven by a sharp analysis of the local linear convergence rate depending on the "active set" at the current iterate. We show that nested-forest decomposition of the inactive edges yields a guaranteed local linear convergence rate. Further, we propose a practical greedy heuristic which realizes such nested decompositions and show in several numerical experiments that our reconditioning strategy, when applied to proximal gradient or primal-dual hybrid gradient algorithm, achieves competitive performances. Our results suggest that local convergence analysis can serve as a guideline for selecting variable metrics in proximal algorithms.