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Fast Frank–Wolfe Algorithms with Adaptive Bregman Step-Size for Weakly Convex Functions

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 6.0Download Report PDF
OptimizationFirst-order methodConvex optimizationNonconvex optimization
Abstract:

We propose a Frank–Wolfe (FW) algorithm with an adaptive Bregman step-size strategy for smooth adaptable (also called: relatively smooth) (weakly-) convex functions. This means that the gradient of the objective function is not necessarily Lipschitz continuous, and we only require the smooth adaptable property. Compared to existing FW algorithms, our assumptions are less restrictive. We establish convergence guarantees in various settings, such as sublinear to linear convergence rates, depending on the assumptions for convex and nonconvex objective functions. Assuming that the objective function is weakly convex and satisfies the local quadratic growth condition, we provide both local sublinear and local linear convergence regarding the primal gap. We also propose a variant of the away-step FW algorithm using Bregman distances over polytopes. We establish global faster (up to linear) convergence for convex optimization under the Hölder error bound condition and its local linear convergence for nonconvex optimization under the local quadratic growth condition. Numerical experiments demonstrate that our proposed FW algorithms outperform existing methods.

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This report is AI-GENERATED using Large Language Models and WisPaper (A scholar search engine). It analyzes academic papers' tasks and contributions against retrieved prior work. While this system identifies POTENTIAL overlaps and novel directions, ITS COVERAGE IS NOT EXHAUSTIVE AND JUDGMENTS ARE APPROXIMATE. These results are intended to assist human reviewers and SHOULD NOT be relied upon as a definitive verdict on novelty.
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Overview

Overall Novelty Assessment

The paper proposes a Frank–Wolfe algorithm with adaptive Bregman step-size for smooth adaptable (relatively smooth) weakly convex functions, extending classical FW methods to settings where gradients lack Lipschitz continuity. It resides in the 'Bregman Distance-Based Adaptive Methods' leaf, which contains only one sibling paper among the seven total papers in the taxonomy. This indicates a relatively sparse research direction within the broader Frank–Wolfe landscape, suggesting the combination of Bregman adaptivity and weakly convex objectives is not yet densely explored.

The taxonomy reveals neighboring branches addressing adaptive parameter selection (smoothness parameter adaptation), composite objectives (regularized and DC optimization), and derivative-free variants. The paper's focus on Bregman-based adaptivity distinguishes it from sibling methods that tune smoothness parameters rather than step sizes, and from composite approaches that handle structured nonsmooth terms. The scope notes clarify that methods requiring known Lipschitz constants or using standard Euclidean distances belong elsewhere, positioning this work at the intersection of adaptive tuning and generalized distance measures for weakly convex problems.

Among 18 candidates examined, the first contribution (adaptive Bregman step-size) shows 1 refutable candidate out of 5 examined, while the second contribution (linear convergence under weaker assumptions) has 2 refutable candidates among 10 examined. The third contribution (away-step variant with Bregman distances over polytopes) found no refutable candidates in 3 examined papers. These statistics suggest that the core adaptive Bregman mechanism and convergence guarantees have some prior overlap within the limited search scope, whereas the away-step polytope extension appears less directly anticipated by the examined literature.

Based on the top-18 semantic matches and taxonomy structure, the work appears to occupy a moderately novel position, particularly in combining Bregman adaptivity with weakly convex objectives and polytope constraints. The limited search scope means deeper prior work may exist beyond the examined candidates, especially in adjacent branches like composite optimization or relative smoothness for convex problems. The sparse taxonomy leaf and partial refutation signals suggest incremental advancement over existing adaptive FW methods rather than a fundamentally new paradigm.

Taxonomy

Core-task Taxonomy Papers
7
3
Claimed Contributions
18
Contribution Candidate Papers Compared
3
Refutable Paper

Research Landscape Overview

Core task: Frank-Wolfe optimization for smooth adaptable weakly convex functions. The field of Frank-Wolfe methods has evolved into several distinct branches that address different aspects of constrained optimization. One major branch focuses on adaptive step-size and parameter selection strategies, where algorithms dynamically tune their internal parameters to improve convergence without requiring problem-specific constants. Another branch tackles composite and structured objective functions, handling settings where the objective decomposes into smooth and nonsmooth parts or exhibits special structure like difference-of-convex formulations. A third direction explores derivative-free and stochastic variants, which are essential when gradient information is expensive or unavailable. Finally, methods exploiting relative smoothness for convex problems have emerged, replacing traditional Lipschitz smoothness assumptions with more flexible distance-based measures. Representative works such as Frank-Wolfe Smooth[3] and Adaptive Frank-Wolfe[5] illustrate how these branches refine classical projection-free algorithms for modern problem classes. Within the adaptive step-size branch, a particularly active line of research centers on Bregman distance-based adaptive methods, which generalize Euclidean geometry to accommodate diverse constraint geometries. Adaptive Bregman Frank-Wolfe[0] sits squarely in this cluster, extending adaptive parameter selection to weakly convex objectives using Bregman divergences. This contrasts with Adaptive Frank-Wolfe[5], which also pursues parameter-free schemes but typically assumes stronger convexity or smoothness properties. Meanwhile, neighboring branches like composite objectives (e.g., Conditional Gradient Composite[1]) and structured nonconvex settings (e.g., Structured Nonconvex Frank-Wolfe[6]) address orthogonal challenges—handling nonsmooth regularizers or difference-of-convex structures—rather than adaptive tuning per se. The interplay between these directions raises open questions about whether adaptive Bregman techniques can be combined with composite or stochastic frameworks, and how relative smoothness notions (Adaptive Relative Smooth[7]) might further relax problem assumptions across all branches.

Claimed Contributions

Frank–Wolfe algorithm with adaptive Bregman step-size for smooth adaptable functions

The authors introduce a Frank–Wolfe algorithm that uses an adaptive Bregman step-size strategy designed for smooth adaptable functions, which relaxes the standard Lipschitz continuity assumption on gradients. This algorithm extends FW methods to a broader class of functions by leveraging Bregman distances instead of Euclidean distances.

5 retrieved papers
Can Refute
Linear convergence guarantees under weaker assumptions for convex and nonconvex optimization

The authors establish theoretical convergence guarantees ranging from sublinear to linear rates for both convex functions under the Hölder error bound condition and weakly convex functions under the local quadratic growth condition. These results achieve linear convergence under weaker assumptions than strong convexity.

10 retrieved papers
Can Refute
Away-step Frank–Wolfe variant using Bregman distances over polytopes

The authors develop a variant of the away-step Frank–Wolfe algorithm that incorporates Bregman distances for optimization over polytopes. They establish global faster convergence for convex problems under the Hölder error bound condition and local linear convergence for nonconvex problems under the local quadratic growth condition.

3 retrieved papers

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution

Frank–Wolfe algorithm with adaptive Bregman step-size for smooth adaptable functions

The authors introduce a Frank–Wolfe algorithm that uses an adaptive Bregman step-size strategy designed for smooth adaptable functions, which relaxes the standard Lipschitz continuity assumption on gradients. This algorithm extends FW methods to a broader class of functions by leveraging Bregman distances instead of Euclidean distances.

Contribution

Linear convergence guarantees under weaker assumptions for convex and nonconvex optimization

The authors establish theoretical convergence guarantees ranging from sublinear to linear rates for both convex functions under the Hölder error bound condition and weakly convex functions under the local quadratic growth condition. These results achieve linear convergence under weaker assumptions than strong convexity.

Contribution

Away-step Frank–Wolfe variant using Bregman distances over polytopes

The authors develop a variant of the away-step Frank–Wolfe algorithm that incorporates Bregman distances for optimization over polytopes. They establish global faster convergence for convex problems under the Hölder error bound condition and local linear convergence for nonconvex problems under the local quadratic growth condition.

Table of Contents

Fast Frank–Wolfe Algorithms with Adaptive Bregman Step-Size for Weakly Convex Functions | Novelty Validation