On the Global Linear Convergence of Frank-Wolfe Optimization Variants (1511.05932v1)

Abstract: The Frank-Wolfe (FW) optimization algorithm has lately re-gained popularity thanks in particular to its ability to nicely handle the structured constraints appearing in machine learning applications. However, its convergence rate is known to be slow (sublinear) when the solution lies at the boundary. A simple less-known fix is to add the possibility to take 'away steps' during optimization, an operation that importantly does not require a feasibility oracle. In this paper, we highlight and clarify several variants of the Frank-Wolfe optimization algorithm that have been successfully applied in practice: away-steps FW, pairwise FW, fully-corrective FW and Wolfe's minimum norm point algorithm, and prove for the first time that they all enjoy global linear convergence, under a weaker condition than strong convexity of the objective. The constant in the convergence rate has an elegant interpretation as the product of the (classical) condition number of the function with a novel geometric quantity that plays the role of a 'condition number' of the constraint set. We provide pointers to where these algorithms have made a difference in practice, in particular with the flow polytope, the marginal polytope and the base polytope for submodular optimization.

• The paper establishes global linear convergence for Frank-Wolfe variants by introducing the innovative pyramidal width metric.
• It analyzes variants like away-steps, pairwise, and fully-corrective methods to address the traditional sublinear convergence of standard FW.
• The findings improve both theoretical insights and practical efficiency in constrained convex optimization, benefiting machine learning applications.

An Insightful Overview of Frank-Wolfe Optimization Variants

The paper "On the Global Linear Convergence of Frank-Wolfe Optimization Variants" by Simon Lacoste-Julien and Martin Jaggi explores several variants of the Frank-Wolfe (FW) algorithm, which is pivotal in constrained convex optimization, particularly within the context of machine learning applications. This paper is significant as it provides comprehensive analysis and proofs of global linear convergence for multiple FW variants, thereby extending their applicability and efficiency in practical scenarios where structured constraints are prevalent.

Variants of Frank-Wolfe Algorithm

The Frank-Wolfe algorithm is traditionally preferred for optimization problems over convex sets due to its ability to handle constraints more naturally than gradient projection methods. However, its convergence has been historically sublinear. The authors address this limitation by examining several variants known for their enhanced performance in certain applications:

• Away-Steps Frank-Wolfe (AFW): Enhances standard FW by adding the possibility of retracting previously chosen atoms, yielding better performance by mitigating zig-zagging issues around optimal boundary points.
• Pairwise Frank-Wolfe (PFW): Operates by swapping weights between atoms, thus offering an improved handling of sparsity in solutions.
• Fully-Corrective Frank-Wolfe (FCFW) and Wolfe's Min-Norm Point (MNP) Algorithm: These methods expand active sets by fully re-optimizing over them, which can result in better convergence properties compared to standard FW.

Theoretical Contributions and Results

The paper's core contribution is establishing the first comprehensive global linear convergence results for all these FW variants under a less stringent requirement than strong convexity. Notably, this is achieved through the use of an innovative metric called the 'pyramidal width,' which reflects the geometric properties of the feasible set. The primary results indicate that:

• These FW variants exhibit a linear decrease in error for strongly convex objectives when optimized over polytopes.
• The convergence is independent of the relative boundary location of the true optimum, addressing a common limitation of previous analyses.
• The convergence constant is determined by the product of the classical condition number of the function and the pyramidal width, emphasizing the role of the geometry of constraints.

Practical and Theoretical Implications

The findings have significant implications for applications involving complex constraint sets, such as the flow polytope, marginal polytope, and base polytope in submodular optimization. By guaranteeing linear convergence, these algorithmic enhancements underscore the practical feasibility of FW variants in reducing computational demands while ensuring optimality.

On a theoretical front, the introduction of the pyramidal width opens pathways for further exploration of geometric measures in optimization. It also harmonizes the Frank-Wolfe method's alignment with affine-invariance principles, thereby facilitating its adaptation across diverse problem domains.

Future Developments

Looking ahead, this work lays the groundwork for developing even more refined variants of FW that can transcend current application barriers. Further research might leverage these insights to explore adaptive mechanisms and algorithmic strategies that enhance scalability and efficacy, particularly in high-dimensional and dynamically constrained environments.

In summary, this paper not only advances our understanding of the Frank-Wolfe algorithm's variants but also sets a new standard in convergence analysis that is both robust and broadly applicable, ensuring its continued prominence in optimization and machine learning tasks.