Projection-free optimization via different variants of the Frank-Wolfe (FW)
method has become one of the cornerstones in large scale optimization for
machine learning and computational statistics. Numerous applications within
these fields involve the minimization of functions with self-concordance like
properties. Such generalized self-concordant (GSC) functions do not necessarily
feature a Lipschitz continuous gradient, nor are they strongly convex. Indeed,
in a number of applications, e.g. inverse covariance estimation or
distance-weighted discrimination problems in support vector machines, the loss
is given by a GSC function having unbounded curvature, implying absence of
theoretical guarantees for the existing FW methods. This paper closes this
apparent gap in the literature by developing provably convergent FW algorithms
with standard O(1/k) convergence rate guarantees. If the problem formulation
allows the efficient construction of a local linear minimization oracle, we
develop a FW method with linear convergence rate.

该论文介绍了投影法优化的一些缺陷，提出了一种基于 Frank-Wolfe 方法的更好优化算法，支持大规模优化，并适用于非凸函数。

Frank-Wolfe 算法的广义自共轭分析

Generalized Self-Concordant Analysis of Frank-Wolfe algorithms

Projection-free optimization via different variants of the Frank-Wolfe (FW),
a.k.a. Conditional Gradient method has become one of the cornerstones in
optimization for machine learning since in many cases the linear minimization
oracle is much cheaper to implement than projections and some sparsity needs to
be preserved. In a number of applications, e.g. Poisson inverse problems or
quantum state tomography, the loss is given by a self-concordant (SC) function
having unbounded curvature, implying absence of theoretical guarantees for the
existing FW methods. We use the theory of SC functions to provide a new
adaptive step size for FW methods and prove global convergence rate O(1/k)
after k iterations. If the problem admits a stronger local linear minimization
oracle, we construct a novel FW method with linear convergence rate for SC
functions.

通过理论建立不同变体的 Frank-Wolfe（FW）算法的自适应步长，对一些机器学习及物理学问题，能够得到无需映射和保留稀疏性的优化，且对于具有无限曲率的自共轭函数，也可以获得全局收敛速率为 O (1/k) 或线性收敛速率的新的 FW 方法。