On the Optimal Memorization Power of ReLU Neural Networks

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 $1\le L\le \sqrt{N}$, 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.

References in corpus (6)

• Depth Separation for Neural Networks
• Neural Networks Learning and Memorization with (almost) no Over-Parameterization
• Depth-Width Trade-offs for ReLU Networks via Sharkovsky's Theorem
• An Exponential Improvement on the Memorization Capacity of Deep Threshold Networks
• Size and Depth Separation in Approximating Benign Functions with Neural Networks
• Depth separation beyond radial functions

Cited by in corpus (1)

• Just Least Squares: Binary Compressive Sampling with Low Generative Intrinsic Dimension