Uniform convergence at optimal rate and scale

Noga Alon; Steve Hanneke; Shay Moran; Tom Waknine

Home/Archive/arXiv

arXiv · 2610.24336

Uniform convergence at optimal rate and scale

Noga Alon·Steve Hanneke·Shay Moran·Tom Waknine

View original source ↗Read source document ↗Export metadata ↓

Abstract

Uniform convergence asks how many random samples suffice for empirical averages to approximate expectations simultaneously over an entire class of functions. It is a central concept in probability and a foundation of statistical learning. For binary-valued classes, the Vapnik--Chervonenkis theorem characterizes uniform convergence by finiteness of the VC dimension. For dimension $d$, the sharp worst-case sample complexity for additive error $\varepsilon$ is of order $d/\varepsilon^2$. For real-valued functions, the analogous combinatorial parameter is the fat-shattering dimension, which depends on an approximation scale $γ$. The challenge is to achieve error $γ+\varepsilon$ with order $d/\varepsilon^2$ samples, matching the sample bound for binary classes, where $d$ is now the fat-shattering dimension at scale $γ$. Previous results achieved either the optimal scale or the optimal sample complexity, but not both. Our main result closes this gap, establishing the sharp sample complexity at the optimal scale. Along the way, we use several tools that may be of independent interest. The first concerns $\varepsilon$-nets with margin. A classical $\varepsilon$-net intersects every set of mass at least $\varepsilon$ in a prescribed family; we allow the net to intersect a slightly enlarged version of each such set. Remarkably, this relaxation permits bounds independent of the ambient dimension in natural geometric settings. Second, we derive improved covering bounds in both discrete and real-valued settings: we extend Haussler's covering lemma to partial concept classes and use this extension to obtain $\ell_2$ covers at the optimal scale for real-valued function classes. Third, we obtain a chaining bound at the optimal scale by aligning the covers so that successive approximations become increasingly close to one another.

Explore related subjects

cs.LG stat.ML

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Noga Alon, Steve Hanneke, Shay Moran, Tom Waknine. 2026-10-17. Uniform convergence at optimal rate and scale. https://arxiv.org/abs/2610.24336

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Heterogeneous Sheaf Neural Networks

Heterogeneous graphs, whose nodes and edges can belong to different types and feature spaces, arise in many real-world domains, including biology, recommendation, social networks, and computer systems. Existing heterogeneous graph neural networks typically handle this heterogeneity at the architectural level through relation-specific modules, meta-path machinery or type-aware attention, which often leads to increasingly specialised parameter-heavy designs. In this work, we propose HetSheaf, a framework for learning heterogeneous graphs through cellular sheaves. Instead of encoding heterogeneity solely in the architecture, HetSheaf represents it directly in the underlying data structure by assigning type-aware local feature spaces and learning restriction maps conditioned on node features, node types, and edge types. To support graph-level prediction, we further introduce SheafPool, a universal stalk-space readout that aggregates node representations while being invariant to local changes of basis, thereby making graph classification with sheaf networks well-defined and achieving an F1 Score up to 42 percentage points higher than mean pooling. Across a diverse suite of benchmarks (node classification, link prediction and graph classification). HetSheaf consistently achieves up to 2 percentage points higher performance (up to 94.97% Macro F1 Score on node classification and up to 99.62% on link prediction) on the Heterogeneous Graph Benchmark (HGB) framework against homogeneous (GCN, GAT, GIN, GraphSAGE), heterogeneous (R-GCN, HAT, HGT) and type-agnostic sheaf baselines, while reducing the number of parameters by up to 10$\times$.

cs.LG↗

arXiv2026

Risk-averse learning with delayed feedback

In real-world scenarios, risk-averse learning is valuable for mitigating potential adverse outcomes. However, the delayed feedback makes it challenging to assess and manage risk effectively. In this paper, we investigate risk-averse learning using Conditional Value at Risk (CVaR) as risk measure, while incorporating feedback with random but bounded delays. We develop two risk-averse learning algorithms that rely on one-point and two-point zeroth-order optimization approaches, respectively. The dynamic regrets of the algorithms are analyzed in terms of the cumulative delay and the number of total samplings. In the absence of delay, the regret bounds match the established bounds of zeroth-order stochastic gradient methods for risk-averse learning. Furthermore, the two-point risk-averse learning outperforms the one-point algorithm by achieving a smaller regret bound. We provide numerical experiments on a dynamic pricing problem to demonstrate the performance of the algorithms.

cs.LG↗

arXiv2026

On the Value of Cross-Modal Misalignment in Multimodal Representation Learning

Multimodal representation learning, exemplified by multimodal contrastive learning (MMCL) using image-text pairs, aims to learn powerful representations by aligning cues across modalities. This approach relies on the core assumption that the exemplar image-text pairs constitute two representations of an identical concept. However, recent research has revealed that real-world datasets often exhibit \emph{cross-modal misalignment}. There are two distinct viewpoints on how to address this issue: one suggests mitigating the misalignment, and the other leveraging it. We seek here to reconcile these seemingly opposing perspectives and to provide a practical guide for practitioners. Using latent variable models, we thus formalize cross-modal misalignment by introducing two specific mechanisms: \emph{selection bias}, where some semantic variables are absent in the text, and \emph{perturbation bias}, where semantic variables are altered---both leading to misalignment in data pairs. Our theoretical analysis shows that, under suitable assumptions, population global minimizers of an alignment--entropy surrogate yield representations that depend only on the semantic variables unaffected by selection and perturbation biases and locally identify this subset almost surely. Extensive empirical studies on synthetic data and real image-text datasets further examine how cross-modal misalignment shapes semantic predictability and downstream performance, shedding light on its nuanced impact on multimodal representation learning.

cs.LG↗