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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 $\tilde{O}(n\barν)$ steps from a warm start in a convex body in $\mathbb{R}^{n}$ using a strongly self-concordant barrier with symmetric self-concordance parameter $\barν$. For many natural barriers, $\barν$ 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 $\tilde{O}(n^{2})$ steps for an arbitrary polytope in $\mathbb{R}^{n}$. 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.