论文标题
$ k $ fw：一种带有更强子问题甲骨文的弗兰克 - 沃尔夫风格算法
$k$FW: A Frank-Wolfe style algorithm with stronger subproblem oracles
论文作者
论文摘要
本文提出了一种新的弗兰克·沃尔夫（FW）的新版本，称为$ k $ fw。标准FW遭受缓慢的收敛性：迭代通常曲折为更新方向，围绕约束集的极端点振荡。新的变体$ k $ fw通过在每次迭代中使用两个更强的子问题甲骨头来克服这个问题。第一个是计算$ k $最佳更新说明（而不是仅一个）的$ K $线性优化Oracle（$ K $ loo）。第二个是$ k $方向搜索（$ k $ ds），该搜索将目标最小化，以$ k $最佳更新方向和以前的迭代为代表的约束集。 When the problem solution admits a sparse representation, both oracles are easy to compute, and $k$FW converges quickly for smooth convex objectives and several interesting constraint sets: $k$FW achieves finite $\frac{4L_f^3D^4}{γδ^2}$ convergence on polytopes and group norm balls, and linear convergence on spectrahedra and nuclear norm balls.数值实验验证了$ k $ fw的有效性，并在现有方法上证明了速度的速度。
This paper proposes a new variant of Frank-Wolfe (FW), called $k$FW. Standard FW suffers from slow convergence: iterates often zig-zag as update directions oscillate around extreme points of the constraint set. The new variant, $k$FW, overcomes this problem by using two stronger subproblem oracles in each iteration. The first is a $k$ linear optimization oracle ($k$LOO) that computes the $k$ best update directions (rather than just one). The second is a $k$ direction search ($k$DS) that minimizes the objective over a constraint set represented by the $k$ best update directions and the previous iterate. When the problem solution admits a sparse representation, both oracles are easy to compute, and $k$FW converges quickly for smooth convex objectives and several interesting constraint sets: $k$FW achieves finite $\frac{4L_f^3D^4}{γδ^2}$ convergence on polytopes and group norm balls, and linear convergence on spectrahedra and nuclear norm balls. Numerical experiments validate the effectiveness of $k$FW and demonstrate an order-of-magnitude speedup over existing approaches.
