  We bound the excess risk of interpolating deep linear networks trained using
gradient flow. In a setting previously used to establish risk bounds for the
minimum $\ell_2$-norm interpolant, we show that randomly initialized deep
linear networks can closely approximate or even match known bounds for the
minimum $\ell_2$-norm interpolant. Our analysis also reveals that interpolating
deep linear models have exactly the same conditional variance as the minimum
$\ell_2$-norm solution. Since the noise affects the excess risk only through
the conditional variance, this implies that depth does not improve the
algorithm's ability to "hide the noise". Our simulations verify that aspects of
our bounds reflect typical behavior for simple data distributions. We also find
that similar phenomena are seen in simulations with ReLU networks, although the
situation there is more nuanced.
