Robust variable structure discovery based on tilted empirical risk minimization

Robust group lasso regression plays an important role in high-dimensional regression modeling such as biological data analysis for disease diagnosis and gene expression. However, most existing methods are optimized with prior variable structure knowledge under the traditional empirical risk minimization (ERM) framework, in which the estimators are excessively dependent on prior structure information and sensitive to outliers. To address this issue, we propose a new robust variable structure discovery method for group lasso based on a convergent bilevel optimization framework. In this paper, we adopt tilted empirical risk minimization (TERM) as the target function to improve the robustness of the estimator by assigning less weight to the outliers. Moreover, we modify the TERM objective function to calculate its Fenchel conjugate while maintaining its robust property, which is proven theoretically and empirically. Experimental results on both synthetic and real-world datasets show that the proposed method can improve the robust performance on prediction and variable structure discovery compared to the existing techniques.