Related papers: Beyond Averaging in John Ellipsoid Approximation: High-Accuracy Algorithms in the Leverage-Score Model

Beyond Averaging in John Ellipsoid Approximation: High-Accuracy Algorithms in the Leverage-Score Model

URL: http://arxiv.org/abs/2606.20082v1
Date: Thu, 18 Jun 2026 11:00:03 GMT
Title: Beyond Averaging in John Ellipsoid Approximation: High-Accuracy Algorithms in the Leverage-Score Model
Abstract summary: John ellipsoid of symmetric polytope $P=mathbfxinmathbbRd:|mathbfAmathbfx|_inftyle1$, $mathbfAinmathbbRntimes d$, is computed by a long line of leverage-score algorithms.
Score: 37.35522173770848
License: http://creativecommons.org/licenses/by/4.0/
Abstract: The John ellipsoid of a symmetric polytope $P=\{\mathbf{x}\in\mathbb{R}^d:\|\mathbf{A}\mathbf{x}\|_\infty\le1\}$, $\mathbf{A}\in\mathbb{R}^{n\times d}$, is computed by a long line of leverage-score algorithms, from Cohen, Cousins, Lee and Yang (COLT 2019) to its successors [WY24, CLS+25], all reaching a $(1+\varepsilon)$-approximation in $Θ(\varepsilon^{-1}\log(n/d))$ iterations. We separate this complexity into three costs the modern line conflates (certification, identification, and accuracy) and locate the historical $\varepsilon^{-1}$ in the first alone. In the equivalent D-optimal-design form $\min_{\mathbf{p}\inΔ_n}-\log\det(\sum_i p_i\mathbf{a}_i\mathbf{a}_i^\top)$, the leverage-score oracle is exactly the first-order oracle and the $(1+\varepsilon)$-John guarantee the Frank-Wolfe gap $g(\mathbf{p})\le\varepsilon d$; through this dictionary the costs come apart. The $\varepsilon^{-1}$ is a certification artifact: the uniform average of the iterates, the certificate used throughout the line, has gap exactly $Θ(1/T)$, however cheap each iteration is made. Pointed instead at the last iterate the same oracle is fast: a warm-started accelerated method reaches the guarantee in $C(\mathbf{A})+O(\sqrtκ\log(1/\varepsilon))$ queries after an $\varepsilon$-independent setup $C(\mathbf{A})$, and once the optimal face is identified the facial problem is an unconstrained self-concordant minimization whose Hessian the oracle recovers exactly, so damped Newton needs only $O(\log\log(1/\varepsilon))$ steps, for a total of $C(\mathbf{A})+O(d^2\log\log(1/\varepsilon))$ queries. The accuracy dependence is thus doubly logarithmic after an $\varepsilon$-independent, condition-dependent setup; the open problem is the remaining identification cost (a condition-free bound on reaching the optimal face) and lower bounds. Accuracy is not the obstruction.

Related papers

A Sharp Barrier for Consistent Submodular Maximization: Any Improvement over $2-\sqrt{2}$ Entails Exponential Queries or Linear Recourse [50.69285844345291]
We study tradeoff between solution quality and stability when elements arrive over time.<n>Our algorithm has a future implementation for rational oracle$ with $O(varepsilon-1)$ recourse and separate the existence of universal-price certificates from their efficient algorithms.
arXiv Detail & Related papers (2026-09-09T10:13:43Z)
Deterministic Coreset for Lp Subspace [10.070877057133004]
We introduce the first iterative algorithm for constructing a $varepsilon$-coreset that guarantees deterministic $ell_p$ subspace embedding.<n>In every iteration, the algorithm ensures that the loss on the maintained set is upper and lower bounded by the loss on the original dataset.<n>Our coreset can also be used for approximately solving the $ell_p$ regression problem in a deterministic manner.
arXiv Detail & Related papers (2026-01-01T14:31:16Z)
Near-Optimal Convergence of Accelerated Gradient Methods under Generalized and $(L_0, L_1)$-Smoothness [57.93371273485736]
We study first-order methods for convex optimization problems with functions $f$ satisfying the recently proposed $ell$-smoothness condition $||nabla2f(x)|| le ellleft(||nabla f(x)||right),$ which generalizes the $L$-smoothness and $(L_0,L_1)$-smoothness.
arXiv Detail & Related papers (2025-08-09T08:28:06Z)
Efficient Continual Finite-Sum Minimization [52.5238287567572]
We propose a key twist into the finite-sum minimization, dubbed as continual finite-sum minimization. Our approach significantly improves upon the $mathcalO(n/epsilon)$ FOs that $mathrmStochasticGradientDescent$ requires. We also prove that there is no natural first-order method with $mathcalOleft(n/epsilonalpharight)$ complexity gradient for $alpha 1/4$, establishing that the first-order complexity of our method is nearly tight.
arXiv Detail & Related papers (2024-06-07T08:26:31Z)
Sample-Efficient Linear Regression with Self-Selection Bias [7.605563562103568]
We consider the problem of linear regression with self-selection bias in the unknown-index setting. We provide a novel and near optimally sample-efficient (in terms of $k$) algorithm to recover $mathbfw_1,ldots,mathbfw_kin. Our algorithm succeeds under significantly relaxed noise assumptions, and therefore also succeeds in the related setting of max-linear regression.
arXiv Detail & Related papers (2024-02-22T02:20:24Z)
Fast swap regret minimization and applications to approximate correlated equilibria [20.047624953965965]
For any constant $varepsilon>0$, our algorithm obtains $varepsilon T$-swap regret within only $T = mathsfpolylog(n)$ rounds. Our algorithm has an exponential dependence on $varepsilon$, but we prove a new, matching lower bound.
arXiv Detail & Related papers (2023-10-30T15:35:24Z)
Fast $(1+\varepsilon)$-Approximation Algorithms for Binary Matrix Factorization [54.29685789885059]
We introduce efficient $(1+varepsilon)$-approximation algorithms for the binary matrix factorization (BMF) problem. The goal is to approximate $mathbfA$ as a product of low-rank factors. Our techniques generalize to other common variants of the BMF problem.
arXiv Detail & Related papers (2023-06-02T18:55:27Z)
Low-degree learning and the metric entropy of polynomials [44.99833362998488]
We prove that any (deterministic or randomized) algorithm which learns $mathscrF_nd$ with $L$-accuracy $varepsilon$ requires at least $Omega(sqrtvarepsilon)2dlog n leq log mathsfM(mathscrF_n,d,|cdot|_L,varepsilon) satisfies the two-sided estimate $$c (1-varepsilon)2dlog
arXiv Detail & Related papers (2022-03-17T23:52:08Z)
Sharper bounds for online learning of smooth functions of a single variable [0.0]
We show that $opt_1+epsilon(mathcalF_q) = Theta(epsilon-frac12)$, where the constants in the bound do not depend on $q$. We also show that $opt_1+epsilon(mathcalF_q) = Theta(epsilon-frac12)$, where the constants in the bound do not depend on $q$.
arXiv Detail & Related papers (2021-05-30T23:06:21Z)
An Optimal Separation of Randomized and Quantum Query Complexity [67.19751155411075]
We prove that for every decision tree, the absolute values of the Fourier coefficients of a given order $ellsqrtbinomdell (1+log n)ell-1,$ sum to at most $cellsqrtbinomdell (1+log n)ell-1,$ where $n$ is the number of variables, $d$ is the tree depth, and $c>0$ is an absolute constant.
arXiv Detail & Related papers (2020-08-24T06:50:57Z)
Robust Gaussian Covariance Estimation in Nearly-Matrix Multiplication Time [14.990725929840892]
We show an algorithm that runs in time $widetildeO(T(N, d) log kappa / mathrmpoly (varepsilon))$, where $T(N, d)$ is the time it takes to multiply a $d times N$ matrix by its transpose. Our runtime matches that of the fastest algorithm for covariance estimation without outliers, up to poly-logarithmic factors.
arXiv Detail & Related papers (2020-06-23T20:21:27Z)
On the Complexity of Minimizing Convex Finite Sums Without Using the Indices of the Individual Functions [62.01594253618911]
We exploit the finite noise structure of finite sums to derive a matching $O(n2)$-upper bound under the global oracle model. Following a similar approach, we propose a novel adaptation of SVRG which is both emphcompatible with oracles, and achieves complexity bounds of $tildeO(n2+nsqrtL/mu)log (1/epsilon)$ and $O(nsqrtL/epsilon)$, for $mu>0$ and $mu=0$
arXiv Detail & Related papers (2020-02-09T03:39:46Z)

This list is automatically generated from the titles and abstracts of the papers in this site.