Uniform Laws of Large Numbers in Product Spaces

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.

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.

We study the fine-grained uniform convergence behavior of halfspaces beyond worst-case VC bounds. For inhomogeneous halfspaces in $\mathbb{R}^d$ with $d\ge 2$, we show that standard first-order VC bounds are essentially tight: even consistent hypotheses can incur population error $Θ(d\ln(n/d)/n)$, and in the agnostic setting the deviation scales as $\sqrt{τ\ln(1/τ)}$ at true error $τ$. In contrast, homogeneous halfspaces in $\mathbb{R}^2$ exhibit a markedly different behavior. In the realizable case, every hypothesis consistent with the sample has error $O(1/n)$. In the agnostic case, we prove a bandwise, log-free deviation bound on each dyadic risk band via a critical-wedge localization argument. Unioning over bands incurs only a $\ln\ln n$ overhead, and we establish a matching lower bound showing this overhead is unavoidable. Together, these results give a fine-grained and nearly complete picture of uniform convergence for halfspaces, revealing sharp dimensional and structural thresholds.