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

Deep Linear Networks can Benignly Overfit when Shallow Ones Do