Log-concave Sampling from a Convex Body with a Barrier: a Robust and Unified Dikin Walk

We consider the problem of sampling from a \(d\)-dimensional log-concave distribution \(\pi(\theta) \propto \exp(-f(\theta))\) for \(L\)-Lipschitz \(f\), constrained to a convex body with an efficiently computable self-concordant barrier function, contained in a ball of radius \(R\) with a \(w\)-warm start. We propose a \emph{robust} sampling framework that computes spectral approximations to the Hessian of the barrier functions in each iteration. We prove that for polytopes that are described by \(n\) hyperplanes, sampling with the Lee-Sidford barrier function mixes within \(\widetilde O((d^2+dL^2R^2)\log(w/\delta))\) steps with a per step cost of \(\widetilde O(nd^{\omega-1})\), where \(\omega\approx 2.37\) is the fast matrix multiplication exponent. Compared to the prior work of Mangoubi and Vishnoi, our approach gives faster mixing time as we are able to design a generalized soft-threshold Dikin walk beyond log-barrier. We further extend our result to show how to sample from a \(d\)-dimensional spectrahedron, the constrained set of a semidefinite program, specified by the set \(\{x\in \mathbb{R}^d: \sum_{i=1}^d x_i A_i \succeq C \}\) where \(A_1,\ldots,A_d, C\) are \(n\times n\) real symmetric matrices. We design a walk that mixes in \(\widetilde O((nd+dL^2R^2)\log(w/\delta))\) steps with a per iteration cost of \(\widetilde O(n^\omega+n^2d^{3\omega-5})\). We improve the mixing time bound of prior best Dikin walk due to Narayanan and Rakhlin that mixes in \(\widetilde O((n^2d^3+n^2dL^2R^2)\log(w/\delta))\) steps.