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版本更新 · v3arXiv:2304.03426

Convex Minimization with Integer Minima in $\widetilde O(n^4)$ Time

Haotian Jiang, Yin Tat Lee, Zhao Song, Lichen Zhang

原文摘要

Given a convex function $f$ on $\mathbb{R}^n$ whose set of minimizers is nonempty and closed, lies in a box of radius $R$ with $\log R\le\mathrm{poly}(n)$, and has integral extreme points, we give a randomized algorithm that finds an integral minimizer of $f$ with high probability using $O(n^2\log n+n\log R)$ calls to a separation oracle, $O(n+\log R/\log n)$ calls to an approximate shortest vector routine such as the LLL algorithm, and $O(n^4\log(nR)\log\log n)$ other arithmetic operations. The previous best polynomial time algorithm [Jiang, SODA 2021, JACM 2022] achieves $O(n(n\log\log n/\log n+\log R))$ oracle complexity, but its overall runtime is at least $\widetildeΩ(n^8)$. We obtain our speedup by combining a variant of the volumetric center cutting plane method of [Vaidya, FOCS 1989], run on the hybrid barrier of [Lee, Sidford, Wong, FOCS 2015] with the fast implementation of [Jiang, Lee, Song, Wong, STOC 2020], and a variant of the LLL algorithm that reduces segments of consecutive basis vectors recursively. The method runs in blocks, so that shortest vectors are computed only $O(n+\log R/\log n)$ times, and each computation takes $O(n^3\log^3 n)$ arithmetic operations in exact arithmetic. For submodular function minimization, our result implies an algorithm using $O(n^3\log n)$ calls to an evaluation oracle and $O(n^4\log^3 n)$ arithmetic operations in total. Its oracle complexity is better than that of the algorithms of [Lee, Sidford, Wong, FOCS 2015] and [Dadush, Végh, Zambelli, SODA 2018, MOR 2021], which had the best running time before this work.

cs.DS · cs.DM · math.OC

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2509.16915

Differential Privacy for Euclidean Jordan Algebra with Applications to Private Symmetric Cone Programming

Zhao Song, Jianfei Xue, Lichen Zhang

原文摘要

In this paper, we study differentially private mechanisms for functions whose outputs lie in a Euclidean Jordan algebra. Euclidean Jordan algebras capture many important mathematical structures and form the foundation of linear programming, second-order cone programming, and semidefinite programming. Our main contribution is a generic Gaussian mechanism for such functions, with sensitivity measured in $\ell_2$, $\ell_1$, and $\ell_\infty$ norms. Notably, this framework includes the important case where the function outputs are symmetric matrices, and sensitivity is measured in the Frobenius, nuclear, or spectral norm. We further derive private algorithms for solving symmetric cone programs under various settings, using a combination of the multiplicative weights update method and our generic Gaussian mechanism. As an application, we present differentially private algorithms for semidefinite programming, resolving a major open question posed by [Hsu, Roth, Roughgarden, and Ullman, ICALP 2014].

math.OC · cs.CR · cs.DS · cs.LG

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2608.26552

Hadamard Flattening and Gaussian Pooling Sketch for Least Squares with Coordinate-wise Guarantee

Zhao Song, Lichen Zhang

原文摘要

Randomized sketch-and-solve algorithms accelerate overconstrained $\ell_2$ regression by replacing the input with a much smaller, randomly projected problem. While standard subspace embeddings nearly preserve the regression cost, coordinate-wise accuracy is more delicate: rather than the objective value, we want the solution vector itself to be close to the optimum in $\ell_\infty$ norm. Specifically, we seek $x'\in\mathbb{R}^d$ with $\|x'-x^\star\|_\infty\leq\fracε{\sqrt d}\|Ax^\star-b\|_2\|A^\dagger\|_{\mathrm{op}}$, where $A\in\mathbb{R}^{n\times d}$ is the design matrix, $b\in\mathbb{R}^n$ the label vector, $x^\star$ the optimal solution and $A^\dagger$ the pseudo-inverse of $A$. Price, Song and Woodruff introduced this problem and claimed that the subsampled randomized Hadamard transform (SRHT) with $O(ε^{-2}d^{1+Θ(\sqrt{\log\log n/\log d})})$ rows achieves this guarantee; we point out a gap in the proof. Song, Ye, Yin and Zhang later claimed $O(ε^{-2}d\log^3 n)$ rows, but their proof relies on an independence assumption that does not hold in general, and we exhibit an explicit instance on which it fails. To achieve a truly nearly-linear-in-$d$ row count, we introduce a new fast, dense randomized transform combining a randomized Hadamard flattening, a random permutation, and balanced, disjoint Gaussian pooling. Conditioned on the Hadamard-and-permutation stage, the sketched problem becomes an exact Gaussian regression whose noise is independent of the entire sketched design, exactly what Song, Ye, Yin and Zhang's argument was missing. Our sketch yields the $\ell_\infty$ guarantee with $m=O(ε^{-2}d\log d)$ rows, using one Hadamard pass with padded internal dimension $N=\widetilde{O}(n+ε^{-2}d^3)$. It is also efficient to apply: $(SA,Sb)$ can be computed in $O(Nd\log N)=\widetilde{O}(nd+ε^{-2}d^4)$ time.

cs.DS · cs.LG · stat.ML

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2509.16801

Sublinear Time Quantum Sensitivity Sampling

Zhao Song, David P. Woodruff, Lichen Zhang

原文摘要

We present a unified quantum sensitivity sampling framework that streamlines coreset construction for many classical approximation problems, significantly speeding up clustering, regression, and low-rank approximation. We assume $n,d\geq\mathrm{poly}(k/ε)$ and, for regression, $n\geq\mathrm{poly}(d)$: - $k$-median/$k$-means: For $n$ points in $\mathbb{R}^d$, we build an $ε$-coreset of input points of size $\tilde O(k^{4/3}ε^{-2})$ (resp. $\tilde O(k^{3/2}ε^{-2})$) in time $\tilde O(ε^{-1}n^{0.5}dk^{1.67})$ (resp. $\tilde O(ε^{-1}n^{0.5}dk^{1.75})$). Our better $d$- and linear $1/ε$-dependence improve on [Xue, Chen, Li and Jiang, ICML'23] for $d\geqΩ(k^{7/3})$ (resp. $Ω(k^{2.5})$). - $\ell_p$ regression: For $\min_x\|Ax-b\|_p$, $A\in\mathbb{R}^{n\times d}$, we build an $ε$-coreset of size $\tilde O_p(d^{\max\{1,p/2\}}ε^{-2})$ in time $\tilde O_p(n^{0.5}d^{1.5}ε^{-1})$ for $p\leq 2$ and $\tilde O_p(n^{0.5}d^{p/4+1}(ε^{-1}+d^{0.5}))$ for $p>2$, matching [Liu and Ji, ICML'26] for $p\leq 2$ and improving on the prior best quantum sampling approach [Apers and Gribling, SICOMP'26] for all $p>2$. - Low-rank approximation: We give the first quantum sublinear-time algorithm for Frobenius-norm rank-$k$ approximation of $A\in\mathbb{R}^{d\times n}$ free of data-dependent parameters, running in $\tilde O(n^{0.5}dk^{0.5}ε^{-0.5})$ time with factored output (plus $\tilde O(nk^2ε^{-2})$ to materialize them). We also give sublinear algorithms for kernel and tensor variants. - Lower bound: For generic sensitivity sampling with total sensitivity $S$, any quantum algorithm outputting an $ε$-coreset of size $o(\sqrt{nS})$ needs $Ω(\sqrt{nS})$ queries, so our query complexity is optimal up to $\mathrm{poly}(ε^{-1},\log n)$ and VC-dimension factors.

cs.DS · cs.LG · quant-ph

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arXiv ↗PDF ↗