  This work studies the (non)robustness of two-layer neural networks in various
high-dimensional linearized regimes. We establish fundamental trade-offs
between memorization and robustness, as measured by the Sobolev-seminorm of the
model w.r.t the data distribution, i.e the square root of the average squared
$L_2$-norm of the gradients of the model w.r.t the its input. More precisely,
if $n$ is the number of training examples, $d$ is the input dimension, and $k$
is the number of hidden neurons in a two-layer neural network, we prove for a
large class of activation functions that, if the model memorizes even a
fraction of the training, then its Sobolev-seminorm is lower-bounded by (i)
$\sqrt{n}$ in case of infinite-width random features (RF) or neural tangent
kernel (NTK) with $d \gtrsim n$; (ii) $\sqrt{n}$ in case of finite-width RF
with proportionate scaling of $d$ and $k$; and (iii) $\sqrt{n/k}$ in case of
finite-width NTK with proportionate scaling of $d$ and $k$. Moreover, all of
these lower-bounds are tight: they are attained by the min-norm / least-squares
interpolator (when $n$, $d$, and $k$ are in the appropriate interpolating
regime). All our results hold as soon as data is log-concave isotropic, and
there is label-noise, i.e the target variable is not a deterministic function
of the data / features. We empirically validate our theoretical results with
experiments. Accidentally, these experiments also reveal for the first time,
(iv) a multiple-descent phenomenon in the robustness of the min-norm
interpolator.
