Parallel Coordinate Descent for L1-Regularized Loss Minimization (1105.5379v1)

Abstract: We propose Shotgun, a parallel coordinate descent algorithm for minimizing L1-regularized losses. Though coordinate descent seems inherently sequential, we prove convergence bounds for Shotgun which predict linear speedups, up to a problem-dependent limit. We present a comprehensive empirical study of Shotgun for Lasso and sparse logistic regression. Our theoretical predictions on the potential for parallelism closely match behavior on real data. Shotgun outperforms other published solvers on a range of large problems, proving to be one of the most scalable algorithms for L1.

• The paper presents the Shotgun algorithm that enables parallel updates in coordinate descent to efficiently minimize L1-regularized losses.
• It establishes theoretical convergence bounds by leveraging the spectral radius, which guides optimal levels of parallelism.
• Empirical results show that Shotgun outperforms traditional methods, particularly in high-dimensional sparse datasets.

Parallel Coordinate Descent for $L_1$-Regularized Loss Minimization

The paper "Parallel Coordinate Descent for $L_1$-Regularized Loss Minimization" presents Shotgun, an innovative parallel coordinate descent algorithm designed for minimizing $L_1$-regularized losses. This paper significantly contributes to the ongoing development of efficient optimization techniques for high-dimensional datasets, particularly in the field of $L_1$-regularization, which is pivotal for promoting sparsity in models.

Overview of Shotgun Algorithm

Traditional coordinate descent has been a preferred method for optimizing $L_1$-regularized models such as the Lasso and sparse logistic regression due to its simplicity and efficiency in high-dimensional settings. However, its inherently sequential nature limits scalability in multi-core environments, which are increasingly prevalent as single-core speeds plateau. The Shotgun algorithm mitigates this limitation by enabling parallel updates of coordinates—the process typically seen as inherently sequential—through careful theoretical groundwork.

Theoretical Contributions

The central theoretical contribution of the work is the derivation of convergence bounds for the Shotgun algorithm, which anticipate linear speedup by parallelizing coordinate updates up to a problem-dependent limit. The authors introduce the concept of spectral radius, $\rho$, of the matrix $A^TA$ (where $A$ is the design matrix), as a key determinant of this limit. The core theorem provides a framework for predicting the maximum number of parallel updates that can be executed without risking divergence—a critical insight for practical implementations.

This analytical approach stands out by not only ensuring convergence but also offering prescriptive insights into the practical estimation of optimal levels of parallelism via easily computable problem characteristics.

Empirical Results

Empirical analysis supports the theoretical claims, demonstrating that Shotgun frequently outperforms existing state-of-the-art solvers across a diverse array of datasets, particularly on large and sparsely featured datasets where traditional methods struggle. Experiments illustrate that the actual speedup in iterations closely aligns with the theoretical predictions, even though runtime speedups are somewhat constrained by hardware limitations, such as memory bandwidth.

The paper also contrasts Shotgun's performance with popular algorithms like SGD and Parallel SGD in the context of sparse logistic regression. The results indicate that while SGD performs efficiently for datasets with a large number of samples relative to features, Shotgun excels in high-dimensional feature spaces, highlighting the potential for hybrid approaches that synthesize gradient descent with coordinate descent techniques.

Practical Implications and Future Directions

The Shotgun algorithm’s scalability makes it especially well-suited for increasingly common scenarios with large feature sets, such as text processing and genomic data. Its robust performance across varied problem instances suggests it is one of the most effective tools currently available for $L_1$-regularized optimization tasks.

Looking forward, an intriguing avenue for future research lies in combining Shotgun's coordinate-based approach with SGD methods, aiming to leverage the strengths of both in mixed-domain applications where datasets simultaneously exhibit large number of features and samples.

In summary, this paper enriches the theoretical and practical landscape of algorithmic optimization by offering a parallel strategy for $L_1$-minimization that promises both theoretical soundness and empirical efficacy. As multi-core infrastructure continues to evolve, algorithms like Shotgun will be crucial in harnessing the full potential of parallel computation for large-scale, high-dimensional data analysis.