Smoothing proximal gradient method for general structured sparse regression (1005.4717v4)

Abstract: We study the problem of estimating high-dimensional regression models regularized by a structured sparsity-inducing penalty that encodes prior structural information on either the input or output variables. We consider two widely adopted types of penalties of this kind as motivating examples: (1) the general overlapping-group-lasso penalty, generalized from the group-lasso penalty; and (2) the graph-guided-fused-lasso penalty, generalized from the fused-lasso penalty. For both types of penalties, due to their nonseparability and nonsmoothness, developing an efficient optimization method remains a challenging problem. In this paper we propose a general optimization approach, the smoothing proximal gradient (SPG) method, which can solve structured sparse regression problems with any smooth convex loss under a wide spectrum of structured sparsity-inducing penalties. Our approach combines a smoothing technique with an effective proximal gradient method. It achieves a convergence rate significantly faster than the standard first-order methods, subgradient methods, and is much more scalable than the most widely used interior-point methods. The efficiency and scalability of our method are demonstrated on both simulation experiments and real genetic data sets.

Smoothing Proximal Gradient Method for General Structured Sparse Regression

The paper by Chen et al. focuses on developing a novel optimization approach for solving high-dimensional sparse regression problems, specifically those that are regularized with structured sparsity-inducing penalties. Such penalties are designed to incorporate prior structural information about the input or output variables, which is advantageous in scenarios where variables have inherent group structures or relational similarities.

Overview of the Methodology

The paper introduces the Smoothing Proximal Gradient (SPG) method, which is tailored for regression problems where the loss function is smooth and convex, and the regularization penalty induces structured sparsity that is both nonseparable and nonsmooth. Two prominent examples of such penalties discussed are the overlapping-group-lasso and the graph-guided-fused-lasso. These are extensions of the traditional Lasso penalty but accommodate more complex relationships among variables by using group-based and graph-based structures, respectively.

1. Smoothing Strategy: The authors employ a smoothing strategy to deal with the nonseparability and nonsmoothness of these penalties. The approach involves reformulating these penalties using dual norms, transforming them into a maximization problem over auxiliary variables. This transformation facilitates the application of Nesterov's smoothing technique to approximate the nonseparable penalties with smooth functions.
2. Optimization Algorithm: The SPG method leverages the Fast Iterative Shrinkage-Thresholding Algorithm (FISTA) framework for optimization. This proximal gradient approach significantly accelerates the convergence compared to standard first-order and subgradient methods and is more scalable than second-order interior-point methods. The algorithm's efficacy is underlined by its convergence rate of $O(1/\varepsilon)$, where $\varepsilon$ denotes the desired accuracy.

Theoretical and Practical Implications

The theoretical contributions of this paper highlight the flexibility and robustness of the SPG method across a general class of structured sparsity-inducing penalties. Importantly, the method is applicable to both single-task and multi-task regression contexts, with smooth, convex loss functions. The implications for practical implementation are significant, offering a scalable solution that can handle large datasets with complex structural sparsity efficiently.

1. Computational Complexity: The paper provides detailed complexity analysis, showcasing that while SPG converges slower than interior-point methods for small-sized problems (due to first-order nature), it becomes advantageous as the problem size increases. This is largely because the per-iteration complexity is substantially lower.
2. Applications: The method was tested on both synthetic and real genetic data, demonstrating its potential in fields like bioinformatics where structured sparsity is commonplace—such as identifying gene interactions in pathways.

Future Directions

The authors speculate on several avenues for future research, such as the exploration of online versions of the algorithm using stochastic gradient descent and incorporation of additional acceleration techniques to further improve performance. These advancements could push the boundaries of current capabilities in large-scale sparse regression analysis.

Overall, the SPG method stands out as a versatile and powerful tool for structured sparse regression, effectively bridging the gap between theoretical robustness and practical applicability across diverse high-dimensional data scenarios. The insights provided by Chen and colleagues pave the way for significant advancements in structured machine learning problems, particularly where underlying relationships between data points can be exploited for improved predictive accuracy.