  Contemporary machine learning applications often involve classification tasks
with many classes. Despite their extensive use, a precise understanding of the
statistical properties and behavior of classification algorithms is still
missing, especially in modern regimes where the number of classes is rather
large. In this paper, we take a step in this direction by providing the first
asymptotically precise analysis of linear multiclass classification. Our
theoretical analysis allows us to precisely characterize how the test error
varies over different training algorithms, data distributions, problem
dimensions as well as number of classes, inter/intra class correlations and
class priors. Specifically, our analysis reveals that the classification
accuracy is highly distribution-dependent with different algorithms achieving
optimal performance for different data distributions and/or training/features
sizes. Unlike linear regression/binary classification, the test error in
multiclass classification relies on intricate functions of the trained model
(e.g., correlation between some of the trained weights) whose asymptotic
behavior is difficult to characterize. This challenge is already present in
simple classifiers, such as those minimizing a square loss. Our novel
theoretical techniques allow us to overcome some of these challenges. The
insights gained may pave the way for a precise understanding of other
classification algorithms beyond those studied in this paper.
