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.

翻译：我们研究了Subramanian等人'22提出的高斯协变量双层模型下，过参数化线性模型在多分类问题中的渐近泛化性。在该模型中，数据点数量、特征维度与类别数量同步增长。我们完全解决了Subramanian等人'22提出的猜想，精确匹配了泛化性的预测区间。此外，我们得到的新下界类似于信息论中的强逆定理：它们确立了误分类率渐近地趋向于0或1。这一紧致结果的一个令人惊讶的推论是：在已知最小范数插值回归器为最优的区间内，最小范数插值分类器相对于非插值分类器可能渐近地处于次优状态。我们紧致分析的关键在于Hanson-Wright不等式的一个新变体，该变体广泛适用于具有稀疏标签的多类问题。作为应用实例，我们证明相同类型的分析可用于分析同一双层集成下的多标签分类相关问题。