# ADMM-SOFTMAX : An ADMM Approach for Multinomial Logistic Regression

## TL;DR

This paper introduces ADMM-Softmax, an efficient ADMM-based algorithm for multinomial logistic regression that improves classification performance and computational efficiency, especially for large-scale image classification tasks.

## Contribution

The paper proposes a novel ADMM-based approach for multinomial logistic regression that decouples the problem into efficiently solvable steps, enabling parallelization and improved generalization.

## Key findings

- ADMM-Softmax outperforms Newton-Krylov, quasi-Newton, and stochastic gradient methods in image classification.
- The method allows for efficient parallelization and pre-computation, reducing computational costs.
- Demonstrates improved generalization on two image classification datasets.

## Abstract

We present ADMM-Softmax, an alternating direction method of multipliers (ADMM) for solving multinomial logistic regression (MLR) problems. Our method is geared toward supervised classification tasks with many examples and features. It decouples the nonlinear optimization problem in MLR into three steps that can be solved efficiently. In particular, each iteration of ADMM-Softmax consists of a linear least-squares problem, a set of independent small-scale smooth, convex problems, and a trivial dual variable update. Solution of the least-squares problem can be be accelerated by pre-computing a factorization or preconditioner, and the separability in the smooth, convex problem can be easily parallelized across examples. For two image classification problems, we demonstrate that ADMM-Softmax leads to improved generalization compared to a Newton-Krylov, a quasi Newton, and a stochastic gradient descent method.