Annealed Sparsity via Adaptive and Dynamic Shrinking

Sparse learning has received tremendous amount of interest in high-dimensional data analysis due to its model interpretability and the low-computational cost. Among the various techniques, adaptive l1-regularization is an effective framework to improve the convergence behaviour of the LASSO, by using varying strength of regularization across different features. In the meantime, the adaptive structure makes it very powerful in modelling grouped sparsity patterns as well, being particularly useful in high-dimensional multi-task problems. However, choosing an appropriate, global regularization weight is still an open problem. In this paper, inspired by the annealing technique in material science, we propose to achieve \"annealed sparsity\" by designing a dynamic shrinking scheme that simultaneously optimizes the regularization weights and model coefficients in sparse (multi-task) learning. The dynamic structures of our algorithm are twofold. Feature-wise (spatially), the regularization weights are updated interactively with model coefficients, allowing us to improve the global regularization structure. Iteration-wise (temporally), such interaction is coupled with gradually boosted l1-regularization by adjusting an equality norm-constraint, achieving an annealing effect to further improve model selection. This renders interesting shrinking behaviour in the whole solution path. Our method competes favorably with state-of-the-art methods in sparse (multi-task) learning. We also apply it in expression quantitative trait loci analysis (eQTL), which gives useful biological insights in human cancer (melanoma) study.