Molecular profiling data (e.g., gene expression) has been used for clinicalrisk prediction and biomarker discovery. However, it is necessary to integrateother prior knowledge like biological pathways or gene interaction networks toimprove the predictive ability and biological interpretability of biomarkers.Here, we first introduce a general regularized Logistic Regression (LR)framework with regularized term $\lambda \|\bm{w}\|_1 +\eta\bm{w}^T\bm{M}\bm{w}$, which can reduce to different penalties, includingLasso, elastic net, and network-regularized terms with different $\bm{M}$. Thisframework can be easily solved in a unified manner by a cyclic coordinatedescent algorithm which can avoid inverse matrix operation and accelerate thecomputing speed. However, if those estimated $\bm{w}_i$ and $\bm{w}_j$ haveopposite signs, then the traditional network-regularized penalty may notperform well. To address it, we introduce a novel network-regularized sparse LRmodel with a new penalty $\lambda \|\bm{w}\|_1 + \eta|\bm{w}|^T\bm{M}|\bm{w}|$to consider the difference between the absolute values of the coefficients. Andwe develop two efficient algorithms to solve it. Finally, we test our methodsand compare them with the related ones using simulated and real data to showtheir efficiency.
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