The optimal information complexity of VC learning

Loss-difference conditional mutual information (ld-CMI) uses the smallest of the standard observations in the supersample hierarchy of generalization bounds: it measures what a learner's loss differences reveal about which candidate of each pair it was trained on. Accuracy is known to force information into the model; data processing does not carry such lower bounds to losses. We show, by bounding three moments of the loss differences, that accuracy also forces ld-CMI. For linear predictors with a smooth convex loss of nonzero slope at zero, such as the logistic loss, plus a regularizer whose curvature and growth are both of power $r\ge2$, on product distributions over a scaled sign cube in dimension at least linear in $n$, every proper learner with expected excess risk at most $\varepsilon$ on these distributions at the optimal sample size $n\asymp\varepsilon^{-2+2/r}$ has worst-case ld-CMI of order $n$ bits, and $Θ(n/(1+(τ/\varepsilon)^2))$ bits under Gaussian noise of standard deviation $τ$ on the loss differences. The same holds without a regularizer, at $n\asymp\varepsilon^{-2}$. Consequently, range-scaled ld-CMI bounds cannot vanish on these distributions, although every proper learner's generalization gap is $O(n^{-1/2})$. We also show that model-level information does not determine noisy loss-difference information, and that the growth, slope and dimension conditions are needed, the last up to a logarithm.

Information theory plays a central role in establishing fundamental limits on what any learning or estimation algorithm can -- and cannot -- achieve, regardless of computational power. In this chapter, we provide an introduction to these connections. End-of-chapter exercises makes the material suitable for both classroom use and self-study. We begin by introducing concentration inequalities along with the notions of covering and packing in metric spaces, and the associated concept of metric entropy. These tools are essential for our analysis. We then introduce the learning-theoretic framework and derive upper bounds on generalization error in terms of metric entropy, Rademacher complexity, and the VC dimension, as well as mutual information and relative entropy. Finally we discuss the minimax estimation framework and establish lower bounds on minimax risk using Fano's inequality, yielding bounds in terms of relative entropy and covering and packing numbers. This manuscript contains preprint of a chapter under consideration for inclusion in the forthcoming third edition of Cover and Thomas's Elements of Information Theory, posted with permission from Wiley. It would follow the chapter posted at arXiv:2605.02989 . The table of contents of the new edition can be found at: https://docs.google.com/document/d/1L-m4oQEJw1PJhoxBeMwrrBD8S_HmvzMEkPbYvS24980/edit?usp=sharing . For feedback, please contact [email protected].

Let $H\subseteq\{-1,+1\}^X$ be a class of finite VC dimension $d\ge1$. Writing $L$ for the binary risk and $L^*=\min_{h\in H}L(h)$, we construct a learner achieving the statistically optimal risk bound: from an i.i.d.\ sample of size $n$, for every $0<δ\le 1/2$, with probability at least $1-δ$, $$L(\widehat h) \le L^*+ 7\cdot10^8\left( \sqrt{\frac{L^*(d+\log(1/δ))}{n}} +\frac{d+\log(1/δ)}{n} \right).$$ This settles the sample complexity of agnostic PAC learning up to universal constants at every fixed $L^*$, matching the lower bounds of Devroye, Györfi, and Lugosi [A Probabilistic Theory of Pattern Recognition, Springer, 1996].