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

• URL: http://arxiv.org/abs/2410.05700v2
• Date: Tue, 12 Nov 2024 19:11:52 GMT
• Title: Log-concave Sampling from a Convex Body with a Barrier: a Robust and Unified Dikin Walk
• Authors: Yuzhou Gu, Nikki Lijing Kuang, Yi-An Ma, Zhao Song, Lichen Zhang,
• Abstract summary: We consider the problem of sampling from a $d$-dimensional log-concave distribution $pi(theta) propto exp(-f(theta))$ for $L$-Lipschitz $f$. We propose a emphrobust sampling framework that computes spectral approximations to the Hessian of the barrier functions in each iteration.
• Score: 12.842909157175582
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• Abstract: 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.

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