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：诸如投影牛顿法、FISTA、镜像下降及其变体等优化算法，享有接近最优的遗憾界和收敛速度，但每次迭代（如在线镜像下降的$O(T^{1/2})$遗憾）中计算“投影”会带来计算瓶颈。另一方面，条件梯度变体每次迭代求解一个线性优化问题，但导致次优的收敛速率（如在线Frank-Wolfe的$O(T^{3/4})$遗憾）。受此运行时间与收敛速率权衡的启发，我们考虑在广泛存在的子模基多面体$B(f)$上对邻近点进行迭代投影。我们首先给出两个邻近点投影到多面体同一面的充要条件，随后证明远离多面体的点以高概率投影到其顶点上。接着，我们利用这一理论，开发一套从离散与连续双重视角加速子模多面体上迭代投影计算的工具集。我们进而调整away-step Frank-Wolfe算法以利用此信息实现提前终止。对于基数型子模多面体的特例，我们将计算特定Bregman投影的运行时间改进$\Omega(n/\log(n))$倍。初步计算实验表明，我们的理论结果可实现运行时间数量级的缩减。