  We initiate the study of the inherent tradeoffs between the size of a neural
network and its robustness, as measured by its Lipschitz constant. We make a
precise conjecture that, for any Lipschitz activation function and for most
datasets, any two-layers neural network with $k$ neurons that perfectly fit the
data must have its Lipschitz constant larger (up to a constant) than
$\sqrt{n/k}$ where $n$ is the number of datapoints. In particular, this
conjecture implies that overparametrization is necessary for robustness, since
it means that one needs roughly one neuron per datapoint to ensure a
$O(1)$-Lipschitz network, while mere data fitting of $d$-dimensional data
requires only one neuron per $d$ datapoints. We prove a weaker version of this
conjecture when the Lipschitz constant is replaced by an upper bound on it
based on the spectral norm of the weight matrix. We also prove the conjecture
in the high-dimensional regime $n \approx d$ (which we also refer to as the
undercomplete case, since only $k \leq d$ is relevant here). Finally we prove
the conjecture for polynomial activation functions of degree $p$ when $n
\approx d^p$. We complement these findings with experimental evidence
supporting the conjecture.
