Sampling from Log-Concave Distributions with Infinity-Distance Guarantees and Applications to Differentially Private Optimization

• URL: http://arxiv.org/abs/2111.04089v1
• Date: Sun, 7 Nov 2021 13:44:50 GMT
• Title: Sampling from Log-Concave Distributions with Infinity-Distance Guarantees and Applications to Differentially Private Optimization
• Authors: Oren Mangoubi and Nisheeth K. Vishnoi
• Abstract summary: We present an algorithm that outputs a point from a distributionO(varepsilon)$close to $$ in infinity-distance. We also present a "soft-pi" version of the Dikin walk which may be independent interest.
• Score: 33.38289436686841
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: For a $d$-dimensional log-concave distribution $\pi(\theta)\propto e^{-f(\theta)}$ on a polytope $K$, we consider the problem of outputting samples from a distribution $\nu$ which is $O(\varepsilon)$-close in infinity-distance $\sup_{\theta\in K}|\log\frac{\nu(\theta)}{\pi(\theta)}|$ to $\pi$. Such samplers with infinity-distance guarantees are specifically desired for differentially private optimization as traditional sampling algorithms which come with total-variation distance or KL divergence bounds are insufficient to guarantee differential privacy. Our main result is an algorithm that outputs a point from a distribution $O(\varepsilon)$-close to $\pi$ in infinity-distance and requires $O((md+dL^2R^2)\times(LR+d\log(\frac{Rd+LRd}{\varepsilon r}))\times md^{\omega-1})$ arithmetic operations, where $f$ is $L$-Lipschitz, $K$ is defined by $m$ inequalities, is contained in a ball of radius $R$ and contains a ball of smaller radius $r$, and $\omega$ is the matrix-multiplication constant. In particular this runtime is logarithmic in $\frac{1}{\varepsilon}$ and significantly improves on prior works. Technically, we depart from the prior works that construct Markov chains on a $\frac{1}{\varepsilon^2}$-discretization of $K$ to achieve a sample with $O(\varepsilon)$ infinity-distance error, and present a method to convert continuous samples from $K$ with total-variation bounds to samples with infinity bounds. To achieve improved dependence on $d$, we present a "soft-threshold" version of the Dikin walk which may be of independent interest. Plugging our algorithm into the framework of the exponential mechanism yields similar improvements in the running time of $\varepsilon$-pure differentially private algorithms for optimization problems such as empirical risk minimization of Lipschitz-convex functions and low-rank approximation, while still achieving the tightest known utility bounds.

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