An Away-Step Frank-Wolfe Method for Minimizing Logarithmically-Homogeneous Barriers

We present and analyze an away-step Frank-Wolfe method for the convex optimization problem ${\min}_{x\in\mathcal{X}} \; f(\mathsf{A} x) + \langle{c},{x}\rangle$, where $f$ is a $\theta$-logarithmically-homogeneous self-concordant barrier, $\mathsf{A}$ is a linear operator that may be non-invertible, $\langle{c},{\cdot}\rangle$ is a linear function and $\mathcal{X}$ is a nonempty polytope. The applications of primary interest include D-optimal design, inference of multivariate Hawkes processes, and TV-regularized Poisson image de-blurring. We establish affine-invariant and norm-independent global linear convergence rates of our method, in terms of both the objective gap and the Frank-Wolfe gap. When specialized to the D-optimal design problem, our results settle a question left open since Ahipasaoglu, Sun and Todd (2008). We also show that the iterates generated by our method will land on and remain in a face of $\mathcal{X}$ within a bounded number of iterations, which can lead to improved local linear convergence rates (for both the objective gap and the Frank-Wolfe gap). We conduct numerical experiments on D-optimal design and inference of multivariate Hawkes processes, and our results not only demonstrate the efficiency and effectiveness of our method compared to other principled first-order methods, but also corroborate our theoretical results quite well.