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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | https://arxiv.org/abs/2410.05700 |
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| _version_ | 1866917835234607104 |
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| author | Gu, Yuzhou Kuang, Nikki Lijing Ma, Yi-An Song, Zhao Zhang, Lichen |
| author_facet | Gu, Yuzhou Kuang, Nikki Lijing Ma, Yi-An Song, Zhao Zhang, Lichen |
| contents | We consider the problem of sampling from a $d$-dimensional log-concave distribution $π(θ) \propto \exp(-f(θ))$ 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 $\widetilde O((d^2+dL^2R^2)\log(w/δ))$ steps with a per step cost of $\widetilde O(nd^{ω-1})$, where $ω\approx 2.37$ 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 $\widetilde O((nd+dL^2R^2)\log(w/δ))$ steps with a per iteration cost of $\widetilde O(n^ω+n^2d^{3ω-5})$. We improve the mixing time bound of prior best Dikin walk due to Narayanan and Rakhlin that mixes in $\widetilde O((n^2d^3+n^2dL^2R^2)\log(w/δ))$ steps. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_05700 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Log-concave Sampling from a Convex Body with a Barrier: a Robust and Unified Dikin Walk Gu, Yuzhou Kuang, Nikki Lijing Ma, Yi-An Song, Zhao Zhang, Lichen Data Structures and Algorithms Machine Learning We consider the problem of sampling from a $d$-dimensional log-concave distribution $π(θ) \propto \exp(-f(θ))$ 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 $\widetilde O((d^2+dL^2R^2)\log(w/δ))$ steps with a per step cost of $\widetilde O(nd^{ω-1})$, where $ω\approx 2.37$ 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 $\widetilde O((nd+dL^2R^2)\log(w/δ))$ steps with a per iteration cost of $\widetilde O(n^ω+n^2d^{3ω-5})$. We improve the mixing time bound of prior best Dikin walk due to Narayanan and Rakhlin that mixes in $\widetilde O((n^2d^3+n^2dL^2R^2)\log(w/δ))$ steps. |
| title | Log-concave Sampling from a Convex Body with a Barrier: a Robust and Unified Dikin Walk |
| topic | Data Structures and Algorithms Machine Learning |
| url | https://arxiv.org/abs/2410.05700 |