Algorithmic Separation between Constant-Depth and Logarithmic-Depth Neural Networks
Yunwei Ren, Zihao Wang, Jason D. Lee
Abstract
Despite the empirical advantages of deep networks over shallow ones, theoretical depth separations largely concern approximation power, while algorithmic results are mostly limited to comparisons between two- and three-layer networks. In this work, we prove the first algorithmic separation between constant-depth and logarithmic-depth networks. Specifically, we identify a class of Boolean functions with hierarchically structured Fourier spectra that logarithmic-depth networks can learn efficiently using layerwise coordinate descent by reconstructing the spectra hierarchically and adaptively. We also exhibit a subclass for which every constant-depth, polynomial-width network with sufficiently regular activations and controlled spectral norms must incur constant $L^2$ approximation error under the uniform distribution over the hypercube.
Topics
Classified with taxonomy v2 on Wed, 2 Sept 2026.