Scale-Sensitive Shattering: Learnability and Evaluability at Optimal Scale

• URL: http://arxiv.org/abs/2605.13684v1
• Date: Wed, 13 May 2026 15:41:30 GMT
• Title: Scale-Sensitive Shattering: Learnability and Evaluability at Optimal Scale
• Abstract summary: We study the optimal scale at which real-valued function classes exhibit uniform convergence and learnability.<n>Our main result establishes a scale-sensitive generalization of the fundamental theorem of PAC learning.<n>We also highlight several open questions on quantitative sample complexity and evaluability.
• Score: 54.65053906803857
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We study the optimal scale at which real-valued function classes exhibit uniform convergence and learnability. Our main result establishes a scale-sensitive generalization of the fundamental theorem of PAC learning: for every bounded real-valued class and every $γ>0$, uniform convergence at scale $γ$, agnostic learnability at scale $γ/2$, and finiteness of the fat-shattering dimension at every scale $γ'>γ$ are equivalent. This resolves a question by Anthony and Bartlett (Cambridge Univ. Press 1999) on the precise scales governing learnability, refuting a conjecture attributed there to Phil Long that a multiplicative 2-factor gap is unavoidable, and improves the upper bounds of Bartlett and Long (JCSS 1998), which incur such a loss. The key technical ingredient is a direct bound on empirical $\ell_\infty$ covering numbers, avoiding the standard detour through packing numbers. As a consequence, we obtain sharp asymptotic metric-entropy bounds in terms of the fat-shattering scale $γ$: an $O(\log^2 n)$ bound holds already at scale $γ/2$, while an $O(\log n)$ bound holds at scale $2γ$. We further show that the $O(\log^2 n)$ bound is sometimes tight. These results resolve open questions by Alon et al. (JACM 1997) and Rudelson and Vershynin (Ann. of Math. 2006). As an application, we establish a sharp dichotomy for bounded integral probability metrics: every such IPM is either estimable or cannot be weakly evaluated within any multiplicative factor $c<3$, while $3$-weak evaluability always holds, resolving an open question from Aiyer et al. (ICML 2026). We also highlight several open questions on quantitative sample complexity and evaluability.

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)
• Hardness of High-Dimensional Linear Classification [58.29089693778071]
  We establish new exponential in dimension lower bounds for the Maximum Halfspace Discrepancy problem.<n>Both are fundamental problems in computational geometry and machine learning in their exact and approximate forms.
  arXiv  Detail & Related papers  (2026-03-19T15:53:41Z)
• Beyond likelihood ratio bias: Nested multi-time-scale stochastic approximation for likelihood-free parameter estimation [49.78792404811239]
  We study inference in simulation-based models where the analytical form of the likelihood is unknown.<n>We use a ratio-free nested multi-time-scale approximation (SA) method that simultaneously tracks the score and drives the parameter update.<n>We show that our algorithm can eliminate the original bias $Obig(sqrtfrac1Nbig)$ and accelerate the convergence rate from $Obig(beta_k+sqrtfracalpha_kNbig)$.
  arXiv  Detail & Related papers  (2024-11-20T02:46:15Z)
• Fast Rates for Bandit PAC Multiclass Classification [73.17969992976501]
  We study multiclass PAC learning with bandit feedback, where inputs are classified into one of $K$ possible labels and feedback is limited to whether or not the predicted labels are correct. Our main contribution is in designing a novel learning algorithm for the agnostic $(varepsilon,delta)$PAC version of the problem.
  arXiv  Detail & Related papers  (2024-06-18T08:54:04Z)
• Revisiting Step-Size Assumptions in Stochastic Approximation [1.3654846342364308]
  It is shown for the first time that this assumption is not required for convergence and finer results.<n>Rates of convergence are obtained for the standard algorithm and for estimates obtained via the averaging technique of Polyak and Ruppert.<n>Results from numerical experiments illustrate that $beta_theta$ may be large due to a combination of multiplicative noise and Markovian memory.
  arXiv  Detail & Related papers  (2024-05-28T05:11:05Z)
• Multilayer Correlation Clustering [12.492037397168579]
  We establish Multilayer Correlation Clustering, a novel generalization of Correlation Clustering (Bansal et al., FOCS '02) to the multilayer setting. In this paper, we are given a series of inputs of Correlation Clustering (called layers) over the common set $V$. The goal is then to find a clustering of $V$ that minimizes the $ell_p$-norm ($pgeq 1$) of the disagreements vector.
  arXiv  Detail & Related papers  (2024-04-25T15:25:30Z)
• Prediction, Learning, Uniform Convergence, and Scale-sensitive Dimensions [39.97534972432276]
  We present a new general-purpose algorithm for learning classes of $[0,1]$-valued functions. We prove a general upper bound on the expected absolute error of this algorithm. We show how to apply both packing bounds to obtain improved general bounds on the sample complexity of learning.
  arXiv  Detail & Related papers  (2023-04-21T15:51:35Z)
• Unitarity estimation for quantum channels [7.323367190336826]
  Unitarity estimation is a basic and important problem in quantum device certification and benchmarking. We provide a unified framework for unitarity estimation, which induces ancilla-efficient algorithms. We show that both the $d$-dependence and $epsilon$-dependence of our algorithms are optimal.
  arXiv  Detail & Related papers  (2022-12-19T09:36:33Z)
• A general approximation lower bound in $L^p$ norm, with applications to feed-forward neural networks [0.0]
  We study the fundamental limits to the expressive power of neural networks. We first prove a general lower bound on how well functions in $F$ can be approximated in $Lp(mu)$ norm. We then instantiate this bound to the case where $G$ corresponds to a piecewise-polynomial feed-forward neural network.
  arXiv  Detail & Related papers  (2022-06-09T09:01:05Z)
• 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)
• Model-Free Reinforcement Learning: from Clipped Pseudo-Regret to Sample Complexity [59.34067736545355]
  Given an MDP with $S$ states, $A$ actions, the discount factor $gamma in (0,1)$, and an approximation threshold $epsilon > 0$, we provide a model-free algorithm to learn an $epsilon$-optimal policy. For small enough $epsilon$, we show an improved algorithm with sample complexity.
  arXiv  Detail & Related papers  (2020-06-06T13:34:41Z)
• Curse of Dimensionality on Randomized Smoothing for Certifiable Robustness [151.67113334248464]
  We show that extending the smoothing technique to defend against other attack models can be challenging. We present experimental results on CIFAR to validate our theory.
  arXiv  Detail & Related papers  (2020-02-08T22:02:14Z)

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.