Papers › Reusing Combinatorial Structure: Faster Iterative Projections over Submodular Base Polytopes
Reusing Combinatorial Structure: Faster Iterative Projections over Submodular Base Polytopes
Jai Moondra, Hassan Mortagy, Swati Gupta
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.
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