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The $L_1$-regularized models are widely used for sparse regression orclassification tasks. In this paper, we propose the orthant-wise passivedescent algorithm (OPDA) for optimizing $L_1$-regularized models, as animproved substitute of proximal algorithms, which are the standard tools foroptimizing the models nowadays. OPDA uses a stochastic variance-reducedgradient (SVRG) to initialize the descent direction, then apply a novelalignment operator to encourage each element keeping the same sign after oneiteration of update, so the parameter remains in the same orthant as before. Italso explicitly suppresses the magnitude of each element to impose sparsity.The quasi-Newton update can be utilized to incorporate curvature informationand accelerate the speed. We prove a linear convergence rate for OPDA ongeneral smooth and strongly-convex loss functions. By conducting experiments on$L_1$-regularized logistic regression and convolutional neural networks, weshow that OPDA outperforms state-of-the-art stochastic proximal algorithms,implying a wide range of applications in training sparse models.

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