Strong self-concordance and sampling

Motivated by the Dikin walk, we develop aspects of the interior-point theory for sampling in high dimension. Specifically, we introduce the notions of strong self-concordance and symmetry for a barrier. These properties imply that the Dikin walk defined using a strongly self-concordant barrier with symmetry parameter ν mixes in Õ( n ν) steps from a warm start for a convex body in ℝ n . For many natural barriers, ν is roughly bounded by ν, the standard self-concordance parameter. We also show that these properties hold for the Lee-Sidford barrier. As a consequence, we obtain the first walk that mixes 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.

Authors

• Aditi Laddha
• Yin Tat Lee
• Santosh S. Vempala

Keywords

• sampling
• polytopes
• self-concordance
• markov chains