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 $\Omega(n/\log(n))$. Our theoretical results show orders of magnitude reduction in runtime in preliminary computational experiments.

翻译：最佳化算法, 如预测牛顿法、 FISTA、 镜底及其变体, 享有接近最佳的遗憾度和趋同率, 但是在计算“ 预测” 的计算瓶颈中, 可能每次循环( 例如, $O( T<unk> 1/2 }) 美元对在线镜色下降的遗憾 ) 。 另一方面, 有条件的梯度变方在每次循环中解决线性优化, 但却导致低于最优化率( 例如, $O( T<unk> 3/4 }) 美元对在线 Frank- Wolfe 的遗憾 。 在运行时间 v/ 趋同率中, 我们考虑到计算“ 预测” 的计算“ 预测” 的计算瓶颈。 我们首先给两个近点项目提供必要和充分的条件, 在每个多面图中, 显示离多面项目多面项目项目( $O( TQ) 极差( T*3/4 } ) 极差( ) 弗兰克- Wolf) 的离我们高概率 。 我们接下来使用这个理论, 开发一个工具, 来加速快速地计算 的离离 直数级数级的直数级计算。</s>