Parametric Maxflows for Structured Sparse Learning with Convex Relaxations of Submodular Functions
This work provides a faster computational method for researchers and practitioners in machine learning dealing with structured sparsity, though it is incremental as it builds on existing parametric maxflow techniques.
The paper tackles the problem of efficiently solving proximal problems for structured sparse learning penalties derived from convex relaxations of submodular functions, showing that these can be addressed using parametric maxflow algorithms with a worst-case runtime only a constant factor higher than a single maxflow computation.
The proximal problem for structured penalties obtained via convex relaxations of submodular functions is known to be equivalent to minimizing separable convex functions over the corresponding submodular polyhedra. In this paper, we reveal a comprehensive class of structured penalties for which penalties this problem can be solved via an efficiently solvable class of parametric maxflow optimization. We then show that the parametric maxflow algorithm proposed by Gallo et al. and its variants, which runs, in the worst-case, at the cost of only a constant factor of a single computation of the corresponding maxflow optimization, can be adapted to solve the proximal problems for those penalties. Several existing structured penalties satisfy these conditions; thus, regularized learning with these penalties is solvable quickly using the parametric maxflow algorithm. We also investigate the empirical runtime performance of the proposed framework.