Faster Sampling from Log-Concave Distributions over Polytopes via a Soft-Threshold Dikin Walk

We consider the problem of sampling from a $d$-dimensional log-concave distribution $\pi(\theta) \propto e^{-f(\theta)}$ constrained to a polytope $K$ defined by $m$ inequalities. Our main result is a "soft-threshold'' variant of the Dikin walk Markov chain 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, where $L$ is the Lipschitz-constant of $f$, $K$ is contained in a ball of radius $R$ and contains a ball of smaller radius $r$, and $\omega$ is the matrix-multiplication constant. When a warm start is not available, it implies an improvement of $\tilde{O}(d^{3.5-\omega})$ arithmetic operations on the previous best bound for sampling from $\pi$ within total variation error $\delta$, which was obtained with the hit-and-run algorithm, in the setting where $K$ is a polytope given by $m=O(d)$ inequalities and $LR = O(\sqrt{d})$. When a warm start is available, our algorithm improves by a factor of $d^2$ arithmetic operations on the best previous bound in this setting, which was obtained for a different version of the Dikin walk algorithm. Plugging our Dikin walk Markov chain into the post-processing algorithm of Mangoubi and Vishnoi (2021), we achieve further improvements in the dependence of the running time for the problem of generating samples from $\pi$ with infinity distance bounds in the special case when $K$ is a polytope.