Bridging the Gap Between Approximation and Learning via Optimal Approximation by ReLU MLPs of Maximal Regularity

Abstract: The foundations of deep learning are supported by the seemingly opposing perspectives of approximation or learning theory. The former advocates for large/expressive models that need not generalize, while the latter considers classes that generalize but may be too small/constrained to be universal approximators. Motivated by real-world deep learning implementations that are both expressive and statistically reliable, we ask: "Is there a class of neural networks that is both large enough to be universal but structured enough to generalize?" This paper constructively provides a positive answer to this question by identifying a highly structured class of ReLU multilayer perceptions (MLPs), which are optimal function approximators and are statistically well-behaved. We show that any $(L,\alpha)$-H\"{o}lder function from $[0,1]^d$ to $[-n,n]$ can be approximated to a uniform $\mathcal{O}(1/n)$ error on $[0,1]^d$ with a sparsely connected ReLU MLP with the same H\"{o}lder exponent $\alpha$ and coefficient $L$, of width $\mathcal{O}(dn^{d/\alpha})$, depth $\mathcal{O}(\log(d))$, with $\mathcal{O}(dn^{d/\alpha})$ nonzero parameters, and whose weights and biases take values in ${0,\pm 1/2}$ except in the first and last layers which instead have magnitude at-most $n$. Further, our class of MLPs achieves a near-optimal sample complexity of $\mathcal{O}(\log(N)/\sqrt{N})$ when given $N$ i.i.d. normalized sub-Gaussian training samples. We achieve this through a new construction that perfectly fits together linear pieces using Kuhn triangulations, along with a new proof technique which shows that our construction preserves the regularity of not only the H\"{o}lder functions, but also any uniformly continuous function. Our results imply that neural networks can solve the McShane extension problem on suitable finite sets.