Multi-level Lasso for Sparse Multi-task Regression

We present a flexible formulation for variable selection in multi-task regression to allow for discrepancies in the estimated sparsity patterns accross the multiple tasks, while leveraging the common structure among them. Our approach is based on an intuitive decomposition of the regression coefficients into a product between a component that is common to all tasks and another component that captures task-specificity. This decomposition yields the Multi-level Lasso objective that can be solved efficiently via alternating optimization. The analysis of the “orthonormal design” case reveals some interesting insights on the nature of the shrinkage performed by our method, compared to that of related work. Theoretical guarantees are provided on the consistency of Multi-level Lasso. Simulations and empirical study of micro-array data further demonstrate the value of our framework.