Sampling from Structured Log-Concave Distributions via a Soft-Threshold Dikin Walk

Given a Lipschitz or smooth convex function $f:K \to \mathbb{R}^d$ for a bounded polytope $K:=${ $\theta \in \mathbb{R}^d: A\theta \leq b$}, where $A\in \mathbb{R}^{m\times d}$ and $b \in \mathbb{R}^m$, we consider the problem of sampling from the log-concave distribution $\pi(\theta) \propto e^{-f(\theta)}$ constrained to $K$. Interest in this problem derives from its applications to Bayesian inference and differential privacy. We present a generalization of the Dikin walk to this setting that requires at most $O((md + d L^2 R^2) \times md^{\omega-1} \log(\frac{w}{\delta}))$ arithmetic operations to sample from $\pi$ within error $\delta>0$ in the total variation distance from a $w$-warm start. Here $L$ is the Lipschitz constant of $f$, $K$ is contained in a ball of radius $R$ and contains a ball of smaller radius $r$, and $\omega \approx 2.37$ is the matrix-multiplication constant. This improves on the running time of prior works for a range of structured settings important for the aforementioned inference and privacy applications. Technically, we depart from previous Dikin walks by adding a soft-threshold regularizer derived from the Lipschitz or smoothness properties of $f$ to a barrier function for $K$ that allows our version of the Dikin walk to propose updates that have a high Metropolis acceptance ratio for $f$, while at the same time remaining inside the polytope $K$.