An homotopy method for lp regression provably beyond self-concordance and in input-sparsity time

We consider the problem of linear regression where the ℓ2n norm loss (i.e., the usual least squares loss) is replaced by the ℓpn norm. We show how to solve such problems up to machine precision in Õp(n|1/2 − 1/p|) (dense) matrix-vector products and Õp(1) matrix inversions, or alternatively in Õp(n|1/2 − 1/p|) calls to a (sparse) linear system solver. This improves the state of the art for any p∉{1,2,+∞}. Furthermore we also propose a randomized algorithm solving such problems in input sparsity time, i.e., Õp(N + poly(d)) where N is the size of the input and d is the number of variables. Such a result was only known for p=2. Finally we prove that these results lie outside the scope of the Nesterov-Nemirovski’s theory of interior point methods by showing that any symmetric self-concordant barrier on the ℓpn unit ball has self-concordance parameter Ω(n).