Sharper Bounds for Chebyshev Moment Matching, with Applications

• URL: http://arxiv.org/abs/2408.12385v2
• Date: Sat, 21 Jun 2025 21:53:59 GMT
• Title: Sharper Bounds for Chebyshev Moment Matching, with Applications
• Authors: Cameron Musco, Christopher Musco, Lucas Rosenblatt, Apoorv Vikram Singh,
• Abstract summary: We study the problem of approximately recovering a probability distribution given noisy measurements of its Chebyshev moments.<n>By leveraging a global decay bound on the coefficients in the Chebyshev expansion of any Lipschitz function, we prove that accurate recovery in the Wasserstein distance is possible with more noise than previously known.
• Score: 26.339024618084476
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We study the problem of approximately recovering a probability distribution given noisy measurements of its Chebyshev polynomial moments. This problem arises broadly across algorithms, statistics, and machine learning. By leveraging a global decay bound on the coefficients in the Chebyshev expansion of any Lipschitz function, we sharpen prior work, proving that accurate recovery in the Wasserstein distance is possible with more noise than previously known. Our result immediately yields a number of applications: 1) We give a simple "linear query" algorithm for constructing a differentially private synthetic data distribution with Wasserstein-$1$ error $\tilde{O}(1/n)$ based on a dataset of $n$ points in $[-1,1]$. This bound is optimal up to log factors, and matches a recent result of Boedihardjo, Strohmer, and Vershynin [Probab. Theory. Rel., 2024], which uses a more complex "superregular random walk" method. 2) We give an $\tilde{O}(n^2/\epsilon)$ time algorithm for the linear algebraic problem of estimating the spectral density of an $n\times n$ symmetric matrix up to $\epsilon$ error in the Wasserstein distance. Our result accelerates prior methods from Chen et al. [ICML 2021] and Braverman et al. [STOC 2022]. 3) We tighten an analysis of Vinayak, Kong, Valiant, and Kakade [ICML 2019] on the maximum likelihood estimator for the statistical problem of "Learning Populations of Parameters'', extending the parameter regime in which sample optimal results can be obtained. Beyond these main results, we provide an extension of our bound to estimating distributions in $d > 1$ dimensions. We hope that these bounds will find applications more broadly to problems involving distribution recovery from noisy moment information.

Related papers

• Nearly-Linear Time Private Hypothesis Selection with the Optimal Approximation Factor [7.069470347531414]
  Estimating the density of a distribution from its samples is a fundamental problem in statistics.<n> Hypothesis selection addresses the setting where, in addition to a sample set, we are given $n$ candidate distributions.<n>We propose a differentially private algorithm in the central model that runs in nearly-linear time with respect to the number of hypotheses.
  arXiv  Detail & Related papers  (2025-06-01T20:46:46Z)
• Can SGD Select Good Fishermen? Local Convergence under Self-Selection Biases and Beyond [11.884593048693507]
  We revisit the problem of estimating $k$ linear regressors with selfselection bias in $doperator dimensions. Our main result is a $namepoly(d,k,1/varepsilon) + kO(k)$ time algorithm for this problem.
  arXiv  Detail & Related papers  (2025-04-06T20:59:12Z)
• Polynomial time sampling from log-smooth distributions in fixed dimension under semi-log-concavity of the forward diffusion with application to strongly dissipative distributions [9.48556659249574]
  We provide a sampling algorithm with complexity in fixed dimension.<n>We prove that our algorithm achieves an expected $epsilon$ error in $KL$ divergence.<n>As an application, we derive an exponential complexity improvement for the problem of sampling from an $L$-log-smooth distribution.
  arXiv  Detail & Related papers  (2024-12-31T17:51:39Z)
• Statistical-Computational Trade-offs for Density Estimation [60.81548752871115]
  We show that for a broad class of data structures their bounds cannot be significantly improved. This is a novel emphstatistical-computational trade-off for density estimation.
  arXiv  Detail & Related papers  (2024-10-30T15:03:33Z)
• A Sample Efficient Alternating Minimization-based Algorithm For Robust Phase Retrieval [56.67706781191521]
  In this work, we present a robust phase retrieval problem where the task is to recover an unknown signal. Our proposed oracle avoids the need for computationally spectral descent, using a simple gradient step and outliers.
  arXiv  Detail & Related papers  (2024-09-07T06:37:23Z)
• Provably Efficient Reinforcement Learning with Multinomial Logit Function Approximation [67.8414514524356]
  We study a new class of MDPs that employs multinomial logit (MNL) function approximation to ensure valid probability distributions over the state space.<n>Despite its significant benefits, incorporating the non-linear function raises substantial challenges in both statistical and computational efficiency.<n>We propose an algorithm that achieves a regret of $widetildemathcalO(dH2sqrtK + kappa-1d2H2)$, eliminating the dependence on $kappa-1$ in the dominant term for the first time
  arXiv  Detail & Related papers  (2024-05-27T11:31:54Z)
• Robust Sparse Estimation for Gaussians with Optimal Error under Huber Contamination [42.526664955704746]
  We study sparse estimation tasks in Huber's contamination model with a focus on mean estimation, PCA, and linear regression. For each of these tasks, we give the first sample and computationally efficient robust estimators with optimal error guarantees. At the technical level, we develop a novel multidimensional filtering method in the sparse regime that may find other applications.
  arXiv  Detail & Related papers  (2024-03-15T15:51:27Z)
• Tractable MCMC for Private Learning with Pure and Gaussian Differential Privacy [23.12198546384976]
  Posterior sampling provides $varepsilon$-pure differential privacy guarantees. It does not suffer from potentially unbounded privacy breach introduced by $(varepsilon,delta)$-approximate DP. In practice, however, one needs to apply approximate sampling methods such as Markov chain Monte Carlo.
  arXiv  Detail & Related papers  (2023-10-23T07:54:39Z)
• Moments, Random Walks, and Limits for Spectrum Approximation [40.43008834125277]
  We show that there are distributions on $[-1,1]$ that cannot be approximated to accuracy $epsilon$ in Wasserstein-1 distance. No algorithm can compute an $epsilon$-accurate approximation to the spectrum of a normalized graph adjacency matrix with constant probability.
  arXiv  Detail & Related papers  (2023-07-02T05:03:38Z)
• On the Linear Convergence of Policy Gradient under Hadamard Parameterization [4.182089296199263]
  We study the convergence of deterministic policy gradient under the Hadamard parameterization. We show that the error decreases at an $O(frac1k)$ rate for all the iterations.
  arXiv  Detail & Related papers  (2023-05-31T05:51:15Z)
• Statistical-Computational Tradeoffs in Mixed Sparse Linear Regression [20.00109111254507]
  We show that the problem suffers from a $frackSNR2$-to-$frack2SNR2$ statistical-to-computational gap. We also analyze a simple thresholding algorithm which, outside of the narrow regime where the problem is hard, solves the associated mixed regression detection problem.
  arXiv  Detail & Related papers  (2023-03-03T18:03:49Z)
• Pseudonorm Approachability and Applications to Regret Minimization [73.54127663296906]
  We convert high-dimensional $ell_infty$-approachability problems to low-dimensional pseudonorm approachability problems. We develop an algorithmic theory of pseudonorm approachability, analogous to previous work on approachability for $ell$ and other norms.
  arXiv  Detail & Related papers  (2023-02-03T03:19:14Z)
• Differentially-Private Hierarchical Clustering with Provable Approximation Guarantees [79.59010418610625]
  We study differentially private approximation algorithms for hierarchical clustering. We show strong lower bounds for the problem: that any $epsilon$-DP algorithm must exhibit $O(|V|2/ epsilon)$-additive error for an input dataset. We propose a private $1+o(1)$ approximation algorithm which also recovers the blocks exactly.
  arXiv  Detail & Related papers  (2023-01-31T19:14:30Z)
• Privately Estimating a Gaussian: Efficient, Robust and Optimal [6.901744415870126]
  We give efficient algorithms for privately estimating a Gaussian distribution in both pure and approximate differential privacy (DP) models. In the pure DP setting, we give an efficient algorithm that estimates an unknown $d$-dimensional Gaussian distribution up to an arbitrary tiny total variation error. For the special case of mean estimation, our algorithm achieves the optimal sample complexity of $widetilde O(d)$, improving on a $widetilde O(d1.5)$ bound from prior work.
  arXiv  Detail & Related papers  (2022-12-15T18:27:39Z)
• Near Sample-Optimal Reduction-based Policy Learning for Average Reward MDP [58.13930707612128]
  This work considers the sample complexity of obtaining an $varepsilon$-optimal policy in an average reward Markov Decision Process (AMDP) We prove an upper bound of $widetilde O(H varepsilon-3 ln frac1delta)$ samples per state-action pair, where $H := sp(h*)$ is the span of bias of any optimal policy, $varepsilon$ is the accuracy and $delta$ is the failure probability.
  arXiv  Detail & Related papers  (2022-12-01T15:57:58Z)
• A Law of Robustness beyond Isoperimetry [84.33752026418045]
  We prove a Lipschitzness lower bound $Omega(sqrtn/p)$ of robustness of interpolating neural network parameters on arbitrary distributions. We then show the potential benefit of overparametrization for smooth data when $n=mathrmpoly(d)$. We disprove the potential existence of an $O(1)$-Lipschitz robust interpolating function when $n=exp(omega(d))$.
  arXiv  Detail & Related papers  (2022-02-23T16:10:23Z)
• Hiding Among the Clones: A Simple and Nearly Optimal Analysis of Privacy Amplification by Shuffling [49.43288037509783]
  We show that random shuffling amplifies differential privacy guarantees of locally randomized data. Our result is based on a new approach that is simpler than previous work and extends to approximate differential privacy with nearly the same guarantees.
  arXiv  Detail & Related papers  (2020-12-23T17:07:26Z)
• List-Decodable Mean Estimation in Nearly-PCA Time [50.79691056481693]
  We study the fundamental task of list-decodable mean estimation in high dimensions. Our algorithm runs in time $widetildeO(ndk)$ for all $k = O(sqrtd) cup Omega(d)$, where $n$ is the size of the dataset. A variant of our algorithm has runtime $widetildeO(ndk)$ for all $k$, at the expense of an $O(sqrtlog k)$ factor in the recovery guarantee
  arXiv  Detail & Related papers  (2020-11-19T17:21:37Z)
• Single-Timescale Stochastic Nonconvex-Concave Optimization for Smooth Nonlinear TD Learning [145.54544979467872]
  We propose two single-timescale single-loop algorithms that require only one data point each step. Our results are expressed in a form of simultaneous primal and dual side convergence.
  arXiv  Detail & Related papers  (2020-08-23T20:36:49Z)
• Optimal Robust Linear Regression in Nearly Linear Time [97.11565882347772]
  We study the problem of high-dimensional robust linear regression where a learner is given access to $n$ samples from the generative model $Y = langle X,w* rangle + epsilon$ We propose estimators for this problem under two settings: (i) $X$ is L4-L2 hypercontractive, $mathbbE [XXtop]$ has bounded condition number and $epsilon$ has bounded variance and (ii) $X$ is sub-Gaussian with identity second moment and $epsilon$ is
  arXiv  Detail & Related papers  (2020-07-16T06:44:44Z)
• Stochastic Saddle-Point Optimization for Wasserstein Barycenters [69.68068088508505]
  We consider the populationimation barycenter problem for random probability measures supported on a finite set of points and generated by an online stream of data. We employ the structure of the problem and obtain a convex-concave saddle-point reformulation of this problem. In the setting when the distribution of random probability measures is discrete, we propose an optimization algorithm and estimate its complexity.
  arXiv  Detail & Related papers  (2020-06-11T19:40:38Z)

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

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.