Accelerating Greedy Coordinate Descent Methods

Abstract: We study ways to accelerate greedy coordinate descent in theory and in practice, where "accelerate" refers either to $O(1/k^2)$ convergence in theory, in practice, or both. We introduce and study two algorithms: Accelerated Semi-Greedy Coordinate Descent (ASCD) and Accelerated Greedy Coordinate Descent (AGCD). While ASCD takes greedy steps in the $x$-updates and randomized steps in the $z$-updates, AGCD is a straightforward extension of standard greedy coordinate descent that only takes greedy steps. On the theory side, our main results are for ASCD: we show that ASCD achieves $O(1/k^2)$ convergence, and it also achieves accelerated linear convergence for strongly convex functions. On the empirical side, we observe that both AGCD and ASCD outperform Accelerated Randomized Coordinate Descent on a variety of instances. In particular, we note that AGCD significantly outperforms the other accelerated coordinate descent methods in numerical tests, in spite of a lack of theoretical guarantees for this method. To complement the empirical study of AGCD, we present a Lyapunov energy function argument that points to an explanation for why a direct extension of the acceleration proof for AGCD does not work; and we also introduce a technical condition under which AGCD is guaranteed to have accelerated convergence. Last of all, we confirm that this technical condition holds in our empirical study.

• The paper presents ASCD and AGCD algorithms that accelerate greedy coordinate descent methods for improved convergence rates.
• It uses a hybrid update strategy combining greedy x-updates with randomized z-updates to achieve O(1/k²) convergence for strongly convex functions.
• Empirical results on linear and logistic regression tasks show that AGCD rapidly reduces objective gaps despite lacking complete theoretical guarantees.

Accelerating Greedy Coordinate Descent Methods

Introduction

The paper "Accelerating Greedy Coordinate Descent Methods" introduces two algorithms, Accelerated Semi-Greedy Coordinate Descent (ASCD) and Accelerated Greedy Coordinate Descent (AGCD), with the aim of improving the convergence speed of greedy coordinate descent methods. These methods are evaluated both theoretically and empirically, demonstrating their potential to outperform existing coordinate descent algorithms. Despite the challenges in proving theoretical guarantees for AGCD, empirical results suggest significant practical gains.

Theoretical Framework

The paper addresses the acceleration of greedy coordinate descent methods, targeting a convergence rate of $O(1/k^2)$. ASCD is shown to achieve this theoretical acceleration by taking greedy steps for $x$-updates while using a randomized approach for $z$-updates. The method obtains accelerated linear convergence for strongly convex functions as well.

For AGCD, although empirical results show promising performance without theoretical convergence guarantees, the authors provide insights through a Lyapunov energy function to explain the challenges encountered in proving its acceleration. They also introduce a technical condition suggesting that AGCD can achieve accelerated convergence under certain circumstances, bridging the gap between theoretical analysis and empirical observations.

Algorithmic Implementation

The implementation of ASCD involves a hybrid approach using greedy coordinate selection for $x$-updates and randomized coordinate selection for $z$-updates within an accelerated framework. The algorithm structure relies on maintaining a balance between these two update sequences to leverage the advantages of both greedy and accelerated randomized methods.

In contrast, AGCD simplifies the approach by exclusively using greedy steps for both update sequences, aligning closer with conventional greedy coordinate descent. The authors underscore that, while AGCD lacks a robust theoretical foundation, its empirical performance is bolstered by demonstrating faster convergence across various problem instances.

Empirical Evaluation

The paper includes comprehensive empirical evaluations, comparing ASCD and AGCD against Accelerated Randomized Coordinate Descent (ARCD). Results show significant performance improvements with AGCD outperforming ASCD and ARCD across most test cases involving linear and logistic regression problems, especially on datasets where strong convexity assumptions are locally valid.

These evaluations highlight AGCD's ability to rapidly achieve lower objective gaps, often reaching convergence faster in practice than theoretical analysis would suggest. The empirical data substantiate the advantages conveyed in terms of reduced computation time and improved scalability over traditional methods.

Conclusion

The work on accelerating greedy coordinate descent methods enriches the algorithmic repertoire for solving large-scale optimization problems, particularly in machine learning scenarios demanding efficiency and speed. While ASCD provides a balance between theoretical soundness and empirical efficacy, AGCD stands out as a practical tool despite the lack of theoretical guarantees. Future research could focus on expanding the theoretical underpinnings of AGCD and exploring broader applications where these methods might excel. The methodology and insights presented have significant implications for advancing optimization techniques in sparse and high-dimensional environments.