Robust testing of low-dimensional functions

• URL: http://arxiv.org/abs/2004.11642v2
• Date: Tue, 12 Jan 2021 22:59:17 GMT
• Title: Robust testing of low-dimensional functions
• Authors: Anindya De, Elchanan Mossel and Joe Neeman
• Abstract summary: A recent work of the authors showed that linear $k$-juntas are testable. Following the surge of interest in noise-tolerant property testing, in this paper we prove a noise-tolerant (or robust) version of this result. We obtain a fully noise tolerant tester with query complexity $kO(mathsfpoly(log k/epsilon))$ for the class of intersection of $k$-halfspaces.
• Score: 8.927163098772589
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: A natural problem in high-dimensional inference is to decide if a classifier $f:\mathbb{R}^n \rightarrow \{-1,1\}$ depends on a small number of linear directions of its input data. Call a function $g: \mathbb{R}^n \rightarrow \{-1,1\}$, a linear $k$-junta if it is completely determined by some $k$-dimensional subspace of the input space. A recent work of the authors showed that linear $k$-juntas are testable. Thus there exists an algorithm to distinguish between: 1. $f: \mathbb{R}^n \rightarrow \{-1,1\}$ which is a linear $k$-junta with surface area $s$, 2. $f$ is $\epsilon$-far from any linear $k$-junta with surface area $(1+\epsilon)s$, where the query complexity of the algorithm is independent of the ambient dimension $n$. Following the surge of interest in noise-tolerant property testing, in this paper we prove a noise-tolerant (or robust) version of this result. Namely, we give an algorithm which given any $c>0$, $\epsilon>0$, distinguishes between 1. $f: \mathbb{R}^n \rightarrow \{-1,1\}$ has correlation at least $c$ with some linear $k$-junta with surface area $s$. 2. $f$ has correlation at most $c-\epsilon$ with any linear $k$-junta with surface area at most $s$. The query complexity of our tester is $k^{\mathsf{poly}(s/\epsilon)}$. Using our techniques, we also obtain a fully noise tolerant tester with the same query complexity for any class $\mathcal{C}$ of linear $k$-juntas with surface area bounded by $s$. As a consequence, we obtain a fully noise tolerant tester with query complexity $k^{O(\mathsf{poly}(\log k/\epsilon))}$ for the class of intersection of $k$-halfspaces (for constant $k$) over the Gaussian space. Our query complexity is independent of the ambient dimension $n$. Previously, no non-trivial noise tolerant testers were known even for a single halfspace.

Related papers

• The Communication Complexity of Approximating Matrix Rank [50.6867896228563]
  We show that this problem has randomized communication complexity $Omega(frac1kcdot n2log|mathbbF|)$. As an application, we obtain an $Omega(frac1kcdot n2log|mathbbF|)$ space lower bound for any streaming algorithm with $k$ passes.
  arXiv  Detail & Related papers  (2024-10-26T06:21:42Z)
• Near-Optimal Bounds for Learning Gaussian Halfspaces with Random Classification Noise [50.64137465792738]
  We show that any efficient SQ algorithm for the problem requires sample complexity at least $Omega(d1/2/(maxp, epsilon)2)$. Our lower bound suggests that this quadratic dependence on $1/epsilon$ is inherent for efficient algorithms.
  arXiv  Detail & Related papers  (2023-07-13T18:59:28Z)
• Phase Transitions in the Detection of Correlated Databases [12.010807505655238]
  We study the problem of detecting the correlation between two Gaussian databases $mathsfXinmathbbRntimes d$ and $mathsfYntimes d$, each composed of $n$ users with $d$ features. This problem is relevant in the analysis of social media, computational biology, etc.
  arXiv  Detail & Related papers  (2023-02-07T10:39:44Z)
• Learning a Single Neuron with Adversarial Label Noise via Gradient Descent [50.659479930171585]
  We study a function of the form $mathbfxmapstosigma(mathbfwcdotmathbfx)$ for monotone activations. The goal of the learner is to output a hypothesis vector $mathbfw$ that $F(mathbbw)=C, epsilon$ with high probability.
  arXiv  Detail & Related papers  (2022-06-17T17:55:43Z)
• Optimal SQ Lower Bounds for Learning Halfspaces with Massart Noise [9.378684220920562]
  tightest statistical query (SQ) lower bounds for learnining halfspaces in the presence of Massart noise. We show that for arbitrary $eta in [0,1/2]$ every SQ algorithm achieving misclassification error better than $eta$ requires queries of superpolynomial accuracy.
  arXiv  Detail & Related papers  (2022-01-24T17:33:19Z)
• Active Sampling for Linear Regression Beyond the $\ell_2$ Norm [70.49273459706546]
  We study active sampling algorithms for linear regression, which aim to query only a small number of entries of a target vector. We show that this dependence on $d$ is optimal, up to logarithmic factors. We also provide the first total sensitivity upper bound $O(dmax1,p/2log2 n)$ for loss functions with at most degree $p$ growth.
  arXiv  Detail & Related papers  (2021-11-09T00:20:01Z)
• Ridgeless Interpolation with Shallow ReLU Networks in $1D$ is Nearest Neighbor Curvature Extrapolation and Provably Generalizes on Lipschitz Functions [16.75218291152252]
  We prove a precise geometric description of all one layer ReLU networks $z(x;theta)$ with a single linear unit. We refer to them as ridgeless ReLU interpolants. Our results show that ridgeless ReLU interpolants achieve the best possible generalization for learning $1d$ Lipschitz functions.
  arXiv  Detail & Related papers  (2021-09-27T11:32:24Z)
• Threshold Phenomena in Learning Halfspaces with Massart Noise [56.01192577666607]
  We study the problem of PAC learning halfspaces on $mathbbRd$ with Massart noise under Gaussian marginals. Our results qualitatively characterize the complexity of learning halfspaces in the Massart model.
  arXiv  Detail & Related papers  (2021-08-19T16:16:48Z)
• Learning a Latent Simplex in Input-Sparsity Time [58.30321592603066]
  We consider the problem of learning a latent $k$-vertex simplex $KsubsetmathbbRdtimes n$, given access to $AinmathbbRdtimes n$. We show that the dependence on $k$ in the running time is unnecessary given a natural assumption about the mass of the top $k$ singular values of $A$.
  arXiv  Detail & Related papers  (2021-05-17T16:40:48Z)

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.