  We study the asymptotic generalization of an overparameterized linear model
for multiclass classification under the Gaussian covariates bi-level model
introduced in Subramanian et al.~'22, where the number of data points,
features, and classes all grow together. We fully resolve the conjecture posed
in Subramanian et al.~'22, matching the predicted regimes for generalization.
Furthermore, our new lower bounds are akin to an information-theoretic strong
converse: they establish that the misclassification rate goes to 0 or 1
asymptotically. One surprising consequence of our tight results is that the
min-norm interpolating classifier can be asymptotically suboptimal relative to
noninterpolating classifiers in the regime where the min-norm interpolating
regressor is known to be optimal.
  The key to our tight analysis is a new variant of the Hanson-Wright
inequality which is broadly useful for multiclass problems with sparse labels.
As an application, we show that the same type of analysis can be used to
analyze the related multilabel classification problem under the same bi-level
ensemble.
