  Deep neural networks generalize well despite being exceedingly
overparameterized and being trained without explicit regularization. This
curious phenomenon has inspired extensive research activity in establishing its
statistical principles: Under what conditions is it observed? How do these
depend on the data and on the training algorithm? When does regularization
benefit generalization? While such questions remain wide open for deep neural
nets, recent works have attempted gaining insights by studying simpler, often
linear, models. Our paper contributes to this growing line of work by examining
binary linear classification under a generative Gaussian mixture model.
Motivated by recent results on the implicit bias of gradient descent, we study
both max-margin SVM classifiers (corresponding to logistic loss) and min-norm
interpolating classifiers (corresponding to least-squares loss). First, we
leverage an idea introduced in [V. Muthukumar et al., arXiv:2005.08054, (2020)]
to relate the SVM solution to the min-norm interpolating solution. Second, we
derive novel non-asymptotic bounds on the classification error of the latter.
Combining the two, we present novel sufficient conditions on the covariance
spectrum and on the signal-to-noise ratio (SNR) under which interpolating
estimators achieve asymptotically optimal performance as overparameterization
increases. Interestingly, our results extend to a noisy model with constant
probability noise flips. Contrary to previously studied discriminative data
models, our results emphasize the crucial role of the SNR and its interplay
with the data covariance. Finally, via a combination of analytical arguments
and numerical demonstrations we identify conditions under which the
interpolating estimator performs better than corresponding regularized
estimates.
