# Automatic Response Category Combination in Multinomial Logistic Regression

## TL;DR

This paper introduces a penalized likelihood approach for multinomial logistic regression that automatically combines response categories, improving model interpretability and prediction accuracy.

## Contribution

It develops a novel non-differentiable penalty and an ADMM algorithm for efficient estimation and category combination in multinomial logistic regression.

## Key findings

- Effective category combination achieved in simulations
- Algorithm converges reliably in practice
- Improved prediction performance over traditional methods

## Abstract

We propose a penalized likelihood method that simultaneously fits the multinomial logistic regression model and combines subsets of the response categories. The penalty is non differentiable when pairs of columns in the optimization variable are equal. This encourages pairwise equality of these columns in the estimator, which corresponds to response category combination. We use an alternating direction method of multipliers algorithm to compute the estimator and we discuss the algorithm's convergence. Prediction and model selection are also addressed.