Fast proximal algorithms for Self-concordant function minimization with application to sparse graph selection

Anastasios Kyrillidis, Volkan Cevher

Abstract: The convex ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -regularized log det divergence criterion has been shown to produce theoretically consistent graph learning. However, this objective function is challenging since the ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -regularization is nonsmooth, the log det objective is not globally Lipschitz gradient function, and the problem is high-dimensional. Using the self-concordant property of the objective, we propose a new adaptive step size selection and present the (F)PS ((F)ast Proximal algorithms for Self-concordant functions) algorithmic framework which has linear convergence and exhibits superior empirical results as compared to state-of-the-art first order methods.