Deep Learning as a Convex Paradigm of Computation: Minimizing Circuit Size with ResNets

arXiv:2511.20888v2 Announce Type: replace
Abstract: This paper argues that DNNs implement a computational Occam’s razor — finding the `simplest’ algorithm that fits the data — and that this could explain their incredible and wide-ranging success over more traditional statistical methods. We start with the discovery that the set of real-valued function $f$ that can be $epsilon$-approximated with a binary circuit of size at most $cepsilon^{-gamma}$ becomes convex in the `Harder than Monte Carlo’ (HTMC) regime, when $gamma>2$, allowing for the definition of a HTMC norm on functions. In parallel one can define a complexity measure on the parameters of a ResNets (a weighted $ell_1$ norm of the parameters), which induce a `ResNet norm’ on functions. The HTMC and ResNet norms can then be related by an almost matching sandwich bound. Thus minimizing this ResNet norm is equivalent to finding a circuit that fits the data with an almost minimal number of nodes (within a power of 2 of being optimal). ResNets thus appear as an alternative model for computation of real functions, better adapted to the HTMC regime and its convexity.