Title: Accelerating Cross-Validation in Multinomial Logistic Regression with $\ell_1$-Regularization,
Abstract: We develop an approximate formula for evaluating a cross-validation estimator
of predictive likelihood for multinomial logistic regression regularized by an
$\ell_1$-norm. This allows us to avoid repeated optimizations required for
literally conducting cross-validation; hence, the computational time can be
significantly reduced. The formula is derived through a perturbative approach
employing the largeness of the data size and the model dimensionality. An
extension to the elastic net regularization is also addressed. The usefulness
of the approximate formula is demonstrated on simulated data and the ISOLET
dataset from the UCI machine learning repository.