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 $Ω(\sqrt{N})$ 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.
