Logarithmic Width Suffices for Robust Memorization
This addresses the challenge of designing robust neural networks for machine learning applications, providing theoretical guarantees on memorization under adversarial conditions.
The paper tackles the problem of robust memorization in feedforward ReLU neural networks against adversarial perturbations, showing that width logarithmic in the number of input samples is necessary and sufficient to achieve robust memorization with a radius independent of sample count.
The memorization capacity of neural networks with a given architecture has been thoroughly studied in many works. Specifically, it is well-known that memorizing $N$ samples can be done using a network of constant width, independent of $N$. However, the required constructions are often quite delicate. In this paper, we consider the natural question of how well feedforward ReLU neural networks can memorize robustly, namely while being able to withstand adversarial perturbations of a given radius. We establish both upper and lower bounds on the possible radius for general $l_p$ norms, implying (among other things) that width logarithmic in the number of input samples is necessary and sufficient to achieve robust memorization (with robustness radius independent of $N$).