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- W3212554751 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 $sup_{thetain K}|logfrac{nu(theta)}{pi(theta)}|$ 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 $O((md+dL^2R^2)times(LR+dlog(frac{Rd+LRd}{varepsilon r}))times md^{omega-1})$ 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." @default.
- W3212554751 created "2021-11-22" @default.
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- W3212554751 date "2021-11-08" @default.
- W3212554751 modified "2023-09-27" @default.
- W3212554751 title "Sampling from Log-Concave Distributions with Infinity-Distance Guarantees and Applications to Differentially Private Optimization." @default.
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