Fast convergence of Frank-Wolfe algorithms on polytopes (2406.18789v4)

Abstract: We provide a template to derive convergence rates for the following popular versions of the Frank-Wolfe algorithm on polytopes: vanilla Frank-Wolfe, Frank-Wolfe with away steps, Frank-Wolfe with blended pairwise steps, and Frank-Wolfe with in-face directions. Our template shows how the convergence rates follow from two affine-invariant properties of the problem, namely, error bound and extended curvature. These properties depend solely on the polytope and objective function but not on any affine-dependent object like norms. For each one of the above algorithms, we derive rates of convergence ranging from sublinear to linear depending on the degree of the error bound.

• The paper introduces a unified framework using extended curvature and error bounds to derive convergence rates for various Frank-Wolfe algorithms on polytopes.
• This framework establishes dimension-independent convergence rates ranging from sublinear to linear, providing a spectrum of results based on error bound properties.
• The analysis considers local facial geometry of polytopes, leading to sharper error bounds than methods relying on overall polytope geometry.

Analysis of Convergence Rates in Frank-Wolfe Algorithms on Polytopes

The paper, Fast Convergence of Frank-Wolfe Algorithms on Polytopes, by Elias Wirth, Javier Pe~na, and Sebastian Pokutta addresses a significant aspect of the Frank-Wolfe (FW) algorithm, a pivotal method in constrained optimization particularly advantageous when projections onto feasible sets, namely polytopes, are computationally expensive. This paper focuses on deriving convergence rates of various Frank-Wolfe variants and introduces a unifying framework founded on two critical properties: extended curvature and error bounds. The manuscript outlines how these properties lead to convergence rates for different Frank-Wolfe variants, offering improvements over previous approaches that often relied on less general assumptions.

Key Contributions

1. Unified Convergence Rate Framework: The authors provide a novel framework that yields convergence rates for vanilla FW and its variants with away steps, blended pairwise steps, and in-face directions. This framework is unique as it leverages affine-invariant properties of the optimization problem, avoiding dependencies on specific norm choices.
2. Extended Convergence Spectrum: By utilizing the extended curvature and error bound properties, the paper demonstrates convergence rates ranging from sublinear to linear. These results are significant as they interpolate between the ubiquitous $O(t^{-1})$ rate of FW under minimal assumptions and enhanced linear rates under stronger assumptions. This interpolation is structured on the discrepancy represented by the error bound property, providing more nuanced insights into the FW algorithm's performance.
3. Dimension-Independent Results: The authors establish convergence rates that do not depend on the dimension of the polytope, a departure from previous results that often required dimension-dependent assumptions. This generality broadens the applicability of their findings and mitigates the computational limitations associated with high-dimensional spaces.
4. Facial Geometry Consideration: They refine the understanding of error bounds through geometric properties of polytopes, specifically examining local facial structures. By focusing on the local geometry, they introduce sharper bounds than those relying on the entire polytope's geometry, such as pyramidal width, which are traditionally used.
5. Advancements for Simplex-like Polytopes: The paper extends and assimilates prior work on simplex-like polytopes, enhancing the linear convergence results without enforcing additional, often brittle conditions like strict complementarity.

Implications and Future Directions

The enhanced framework for convergence analysis can significantly impact how conditional gradient methods, notably FW variants, are utilized in practical applications requiring fast and dimension-invariant solutions. The results set a precedent for optimization on complex feasible regions, extending applicability to problems in machine learning, operations research, and large-scale discrete optimizations.

Theoretically, this paper opens pathways for further investigation into the relationship between polytope geometry and optimization dynamics, suggesting that understanding geometric and error-bound properties can be pivotal in designing more efficient algorithms. Practically, the results may inform the development of optimization algorithms in environments constrained by computational resources, where efficient computation of projections is imperative.

In conclusion, this paper reshapes the landscape of convergence analysis for Frank-Wolfe algorithms. By grounding their contributions in robust, invariant properties of polytopes, the authors not only enhance the theoretical understanding but also potentiate the practical utility of Frank-Wolfe methods across a spectrum of applications. Future research may enhance these findings through empirical validation in diverse optimization contexts and adaptation to newer variants and generalizations of Frank-Wolfe and other projection-free algorithms.