Reusing Combinatorial Structure: Faster Iterative Projections over Submodular Base Polytopes

Abstract

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) [1,2,3,4]. 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) [5,6,7,8]. 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 develop a toolkit to speed up the computation of projections using both discrete and continuous perspectives (e.g., [9,10,11]). 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.