Reusing Combinatorial Structure: Faster Iterative Projections over Submodular Base Polytopes
This addresses a key efficiency problem for optimization practitioners, offering incremental improvements in projection speed for specific polytope types.
The paper tackles the computational bottleneck of iterative projections in optimization algorithms by developing a method to speed up projections over submodular base polytopes, resulting in a runtime improvement by a factor of Ω(n/log(n)) for cardinality-based cases and showing orders of magnitude reduction in experiments.
Optimization algorithms such as projected Newton's method, FISTA, mirror descent, and its variants enjoy near-optimal regret bounds and convergence rates, but suffer from a computational bottleneck of computing ``projections'' in potentially each iteration (e.g., $O(T^{1/2})$ regret of online mirror descent). On the other hand, conditional gradient variants solve a linear optimization in each iteration, but result in suboptimal rates (e.g., $O(T^{3/4})$ regret of online Frank-Wolfe). Motivated by this trade-off in runtime v/s convergence rates, we consider iterative projections of close-by points over widely-prevalent submodular base polytopes $B(f)$. We first give necessary and sufficient conditions for when two close points project to the same face of a polytope, and then show that points far away from the polytope project onto its vertices with high probability. We next use this theory and develop a toolkit to speed up the computation of iterative projections over submodular polytopes using both discrete and continuous perspectives. We subsequently adapt the away-step Frank-Wolfe algorithm to use this information and enable early termination. For the special case of cardinality-based submodular polytopes, we improve the runtime of computing certain Bregman projections by a factor of $Ω(n/\log(n))$. Our theoretical results show orders of magnitude reduction in runtime in preliminary computational experiments.