Reusing Combinatorial Structure: Faster Iterative Projections over Submodular Base Polytopes
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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 develop a toolkit to speed up the computation of projections 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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