Enhancing Parameter-Free Frank Wolfe with an Extra Subproblem

TL;DR

This paper introduces ExtraFW, a parameter-free Frank Wolfe variant with a prediction-correction update, achieving optimal convergence rates and superior empirical performance in machine learning tasks like classification and matrix completion.

Contribution

The paper proposes ExtraFW, a novel parameter-free Frank Wolfe algorithm with a prediction-correction step, offering improved convergence rates and empirical results over existing methods.

Findings

Achieves ${ m O}(1/k)$ convergence rate for general convex problems.

Attains ${ m O}(1/k^2)$ rate on certain machine learning problems.

Outperforms standard FW and Nesterov's accelerated gradient in experiments.

Abstract

Aiming at convex optimization under structural constraints, this work introduces and analyzes a variant of the Frank Wolfe (FW) algorithm termed ExtraFW. The distinct feature of ExtraFW is the pair of gradients leveraged per iteration, thanks to which the decision variable is updated in a prediction-correction (PC) format. Relying on no problem dependent parameters in the step sizes, the convergence rate of ExtraFW for general convex problems is shown to be ${\cal O}(\frac{1}{k})$, which is optimal in the sense of matching the lower bound on the number of solved FW subproblems. However, the merit of ExtraFW is its faster rate ${\cal O}\big(\frac{1}{k^2} \big)$ on a class of machine learning problems. Compared with other parameter-free FW variants that have faster rates on the same problems, ExtraFW has improved rates and fine-grained analysis thanks to its PC update. Numerical tests on binary classification with different sparsity-promoting constraints demonstrate that the empirical performance of ExtraFW is significantly better than FW, and even faster than Nesterov's accelerated gradient on certain datasets. For matrix completion, ExtraFW enjoys smaller optimality gap, and lower rank than FW.