[Paper Review] Tree-Guided Group Lasso for Multi-Task Regression with Structured Sparsity

Se Young Kim, Eric P. Xing|arXiv (Cornell University)|Jun 29, 2018

Statistical Methods and Inference16 citations

TL;DR

This paper proposes tree-guided group lasso, a structured regularization method for multi-task regression that leverages a priori tree-structured output relationships to enforce sparsity across clusters of outputs. By designing a systematic weighting scheme for overlapping groups in the group lasso penalty, the method ensures balanced penalization and enables efficient large-scale optimization, demonstrating superior prediction accuracy and sparsity pattern recovery on simulated and yeast datasets compared to existing methods.

ABSTRACT

We consider the problem of learning a sparse multi-task regression, where the structure in the outputs can be represented as a tree with leaf nodes as outputs and internal nodes as clusters of the outputs at multiple granularity. Our goal is to recover the common set of relevant inputs for each output cluster. Assuming that the tree structure is available as prior knowledge, we formulate this problem as a new multi-task regularized regression called tree-guided group lasso. Our structured regularization is based on a grouplasso penalty, where groups are defined with respect to the tree structure. We describe a systematic weighting scheme for the groups in the penalty such that each output variable is penalized in a balanced manner even if the groups overlap. We present an efficient optimization method that can handle a largescale problem. Using simulated and yeast datasets, we demonstrate that our method shows a superior performance in terms of both prediction errors and recovery of true sparsity patterns compared to other methods for multi-task learning.

Motivation & Objective

• To address multi-task regression with structured output relationships by incorporating prior tree knowledge into the learning process.
• To develop a regularization method that promotes sparsity at multiple levels of output clustering defined by a tree structure.
• To ensure balanced penalization of individual outputs despite overlapping groups in the tree-structured hierarchy.
• To enable efficient optimization for large-scale multi-task regression problems with structured sparsity.
• To improve both prediction accuracy and recovery of true underlying sparsity patterns in multi-task learning.

Proposed method

• The method formulates multi-task regression using a group lasso penalty where groups are defined by internal nodes of a tree structure, with leaves representing individual outputs.
• A systematic weighting scheme is introduced to balance the penalization of each output across overlapping groups, ensuring fairness and consistency in sparsity induction.
• The regularization penalty is constructed such that each output is penalized proportionally to its membership in multiple hierarchical groups, preventing bias toward deeper or shallower nodes.
• An efficient optimization algorithm is developed to solve the large-scale problem, leveraging proximal methods tailored to the tree-structured group lasso formulation.
• The method integrates prior knowledge of output relationships via a tree, allowing for hierarchical clustering of outputs and shared feature selection across clusters.
• The optimization process ensures convergence and scalability, making the method applicable to high-dimensional, large-scale datasets.

Experimental results

Research questions

• RQ1 Can incorporating tree-structured output relationships improve the accuracy and sparsity recovery in multi-task regression?
• RQ2 How can overlapping groups in a hierarchical tree structure be penalized fairly to avoid bias in feature selection?
• RQ3 Does the proposed tree-guided group lasso outperform standard group lasso and other multi-task learning methods in terms of prediction error and sparsity pattern recovery?
• RQ4 Can the method scale efficiently to large-scale datasets while maintaining structured sparsity?
• RQ5 To what extent does the method recover the true underlying sparsity patterns in synthetic and real-world data?

Key findings

• The tree-guided group lasso method achieves significantly lower prediction errors compared to baseline methods on both simulated and real yeast datasets.
• The method demonstrates superior recovery of true sparsity patterns, particularly in capturing hierarchical relationships among outputs.
• The systematic weighting scheme effectively balances penalization across overlapping groups, preventing bias toward certain nodes in the tree structure.
• The optimization algorithm efficiently handles large-scale problems, enabling application to high-dimensional datasets with thousands of outputs.
• Empirical results on yeast data confirm that the method identifies relevant input features shared across biologically meaningful output clusters.
• The method outperforms standard group lasso and other multi-task learning baselines in both predictive performance and sparsity pattern accuracy.