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 symmetric and 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 Rn 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(n2) steps for an arbitrary polytope in Rⁿ. 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.