Strong self-concordance and sampling

Motivated by the Dikin walk, we develop aspects of an interior-point theory for sampling in high dimension. Specifically, we introduce a symmetric parameter and the notion of strong self-concordance. These properties imply that the corresponding Dikin walk mixes in Õ(nν) steps from a warm start in a convex body in R n using a strongly self-concordant barrier with symmetric self-concordance parameter ν. For many natural barriers, ν is roughly bounded by ν, the standard self-concordance parameter. We show that this property and strong self-concordance hold for the Lee-Sidford barrier. As a consequence, we obtain the first walk to mix in Õ(n 2 ) steps for an arbitrary polytope in R n . Strong self-concordance for other barriers leads to an interesting (and unexpected) connection -for the universal and entropic barriers, it is implied by the KLS conjecture.