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 tildeO(nbarnu) steps from a warm start in a convex body in ℝn using a strongly self-concordant barrier with symmetric self-concordance parameter barnu. For many natural barriers, barnu is roughly bounded by nu, 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 tildeO(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.