Self-Concordant Analysis of Frank-Wolfe Algorithms
This work addresses a bottleneck in projection-free optimization for applications like Poisson inverse problems or quantum state tomography, offering incremental improvements with new convergence guarantees.
The paper tackles the problem of optimizing self-concordant functions with unbounded curvature, which lacks theoretical guarantees in existing Frank-Wolfe methods, by introducing a new adaptive step size that achieves a global convergence rate of O(1/k) and a novel method with linear convergence under stronger conditions.
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.