Multi-Task Feature Learning Via Efficient l2,1-Norm Minimization

TLDR

The problem of joint feature selection across related tasks arises in biomedical informatics and computer vision, and the l2,1‑norm regularized regression model encourages shared sparsity patterns but leads to a challenging non‑smooth optimization problem. In this paper, we accelerate the computation by reformulating the problem as two equivalent smooth convex optimization problems that are solved via Nesterov’s optimal first‑order method. A key building block in solving these reformulations is the Euclidean projection. The Euclidean projection for the first reformulation can be analytically computed, while for the second it can be computed in linear time, and empirical evaluations on several data sets confirm the efficiency of the proposed algorithms.

Abstract

The problem of joint feature selection across a group of related tasks has applications in many areas including biomedical informatics and computer vision. We consider the l2,1-norm regularized regression model for joint feature selection from multiple tasks, which can be derived in the probabilistic framework by assuming a suitable prior from the exponential family. One appealing feature of the l2,1-norm regularization is that it encourages multiple predictors to share similar sparsity patterns. However, the resulting optimization problem is challenging to solve due to the non-smoothness of the l2,1-norm regularization. In this paper, we propose to accelerate the computation by reformulating it as two equivalent smooth convex optimization problems which are then solved via the Nesterov's method-an optimal first-order black-box method for smooth convex optimization. A key building block in solving the reformulations is the Euclidean projection. We show that the Euclidean projection for the first reformulation can be analytically computed, while the Euclidean projection for the second one can be computed in linear time. Empirical evaluations on several data sets verify the efficiency of the proposed algorithms.