Reusing Combinatorial Structure: Faster Iterative Projections over Submodular Base Polytopes

Jai Moondra, Hassan Mortagy, Swati Gupta

Keywords: Convex optimization, Conditional Gradients, Bregman Projections, Submodular base polytopes, Online learning

TL;DR: We bridge discrete and continuous optimization approaches to speed up iterative Bregman 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). 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 $\Omega(n/\log(n))$. Our theoretical results show orders of magnitude reduction in runtime in preliminary computational experiments.

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Code: https://github.com/jaimoondra/submodular-polytope-projections