On the Optimal Memorization Power of ReLU Neural Networks

Gal Vardi, Gilad Yehudai, Ohad Shamir

Keywords: Expressivness, Memorization, Theory, VC-dimension, Deep learning theory

Abstract: We study the memorization power of feedforward ReLU neural networks. We show that such networks can memorize any $N$ points that satisfy a mild separability assumption using $\tilde{O}\left(\sqrt{N}\right)$ parameters. Known VC-dimension upper bounds imply that memorizing $N$ samples requires $\Omega(\sqrt{N})$ parameters, and hence our construction is optimal up to logarithmic factors. We also give a generalized construction for networks with depth bounded by $1 \leq L \leq \sqrt{N}$, for memorizing $N$ samples using $\tilde{O}(N/L)$ parameters. This bound is also optimal up to logarithmic factors. Our construction uses weights with large bit complexity. We prove that having such a large bit complexity is both necessary and sufficient for memorization with a sub-linear number of parameters.

One-sentence Summary: We show that ReLU neural networks can memorize N samples using \sqrt{N} parameters, and prove that up to logarithmic terms this is the optimal solution.