The Frank-Wolfe method (a.k.a. conditional gradient algorithm) for smooth optimization has regained much interest in recent years in the context of large scale optimization and machine learning. A key advantage of the method is that it avoids projections - the computational bottleneck in many applications - replacing it by a linear optimization step. Despite this advantage, the convergence rates of the FW method fall behind standard gradient methods for most settings of interest. It is an active line of research to derive faster FW algorithms for various settings of convex optimization. In this paper we consider the special case of optimization over strongly convex sets, for which we prove that the vanila FW method converges at a rate of $\frac{1}{t^2}$. This gives a quadratic improvement in convergence rate compared to the general case, in which convergence is of the order $\frac{1}{t}$, and known to be tight. We show that various balls induced by $p$-norms, Schatten norms and group norms are strongly convex on one hand and on the other hand, linear optimization over these sets is straightforward and admits a closed-form solution. This gives a family of applications, e.g. linear regression with regularization, for which the FW algorithm enjoys both worlds: a fast convergence rate and no projections.

在本篇论文中，我们考虑了在强凸集上进行的优化的特殊情况。我们证明，与一般情况的收敛速度为1/t相比，vanila FW方法以1/t²的速度收敛。我们还展示了如何通过在这些集合上进行线性优化来推导FW方法的多个快速收敛结果。

Frank-Wolfe方法在强凸集合上的更快收敛速率