A fast and efficient smoothing approach to Lasso regression and an application in statistical genetics: polygenic risk scores for chronic obstructive pulmonary disease (COPD)

Abstract

High dimensional linear regression problems are often fitted using Lasso approaches. Although the Lasso objective function is convex, it is not differentiable everywhere, making the use of gradient descent methods for minimization not straightforward. To avoid this technical issue, we apply Nesterov smoothing to the original (unsmoothed) Lasso objective function. We introduce a closed-form smoothed Lasso which preserves the convexity of the Lasso function, is uniformly close to the unsmoothed Lasso, and allows us to obtain closed-form derivatives everywhere for efficient and fast minimization via gradient descent. Our simulation studies are focused on polygenic risk scores using genetic data from a genome-wide association study (GWAS) for chronic obstructive pulmonary disease (COPD). We compare accuracy and run-time of our approach to the current gold standard in the literature, the FISTA algorithm. Our results suggest that the proposed methodology provides estimates with equal or higher accuracy than the FISTA algorithm while having the same asymptotic runtime scaling. The proposed methodology is implemented in the R-package smoothedLasso, available on the Comprehensive R Archive Network (CRAN).