# Training L1-Regularized Models with Orthant-Wise Passive Descent Algorithms

## TL;DR

This paper introduces OPDA, a novel orthant-wise passive descent algorithm that improves the optimization of L1-regularized models by maintaining parameter signs and promoting sparsity, with proven linear convergence.

## Contribution

The paper proposes OPDA, a new algorithm combining SVRG, an alignment operator, and quasi-Newton updates for efficient L1-regularized model training, outperforming existing methods.

## Key findings

- OPDA achieves faster convergence than state-of-the-art stochastic proximal algorithms.
- OPDA effectively maintains parameter orthants and promotes sparsity.
- Experimental results show OPDA's superior performance on logistic regression and CNNs.

## Abstract

The $L_1$-regularized models are widely used for sparse regression or classification tasks. In this paper, we propose the orthant-wise passive descent algorithm (OPDA) for optimizing $L_1$-regularized models, as an improved substitute of proximal algorithms, which are the standard tools for optimizing the models nowadays. OPDA uses a stochastic variance-reduced gradient (SVRG) to initialize the descent direction, then apply a novel alignment operator to encourage each element keeping the same sign after one iteration of update, so the parameter remains in the same orthant as before. It also explicitly suppresses the magnitude of each element to impose sparsity. The quasi-Newton update can be utilized to incorporate curvature information and accelerate the speed. We prove a linear convergence rate for OPDA on general smooth and strongly-convex loss functions. By conducting experiments on $L_1$-regularized logistic regression and convolutional neural networks, we show that OPDA outperforms state-of-the-art stochastic proximal algorithms, implying a wide range of applications in training sparse models.