New Projection-Free Algorithm BAGEL Achieves Optimal Regret in Online Convex Optimization

Global: New Projection-Free Algorithm BAGEL Achieves Optimal Regret in Online Convex Optimization

Researchers Yiyang Lu, Mohammad Pedramfar, and Vaneet Aggarwal announced on arXiv that their algorithm named BAGEL offers a projection‑free solution for adversarially constrained online convex optimization (COCO), achieving optimal (O(T^{1/2})) regret while maintaining low cumulative constraint violation. The work was first submitted on 23 February 2025 and revised on 28 January 2026, aiming to improve scalability of online learning methods without sacrificing theoretical performance.

Background

Constrained online convex optimization requires an algorithm to make sequential decisions within a feasible set while minimizing cumulative loss. Traditional projection‑based approaches guarantee (O(T^{1/2})) regret but rely on costly projection operations that can hinder performance on high‑dimensional or complex constraint sets.

Limitations of Existing Projection‑Free Methods

Alternative projection‑free strategies replace projections with linear optimization oracles (LOOs), which are computationally cheaper. However, these methods typically attain slower regret rates of (O(T^{3/4})), creating a trade‑off between efficiency and theoretical guarantees.

Introducing BAGEL

The BAGEL algorithm leverages a stronger oracle— a Separation Oracle (SO)—to navigate the feasible region without explicit projections. By employing an infeasible projection technique based on the SO, BAGEL recovers the (O(T^{1/2})) regret bound and attains a (tilde{O}(T^{1/2})) bound on cumulative constraint violation for convex cost functions.

Performance Guarantees

Analysis shows that BAGEL matches the time‑horizon dependence of projection‑based algorithms while requiring only (tilde{O}(T)) oracle calls. The regret and violation bounds hold under standard convexity assumptions, with the algorithm’s complexity scaling with geometric properties of the action set rather than dimensionality.

Implications and Future Directions

The results establish a specific regime where projection‑free methods can achieve the same convergence rates as their projection‑based counterparts, potentially enabling more scalable online learning systems in large‑scale applications. The authors suggest that further investigation into oracle design and adaptive geometry handling could broaden BAGEL’s applicability across diverse optimization scenarios.

This report is based on information from arXiv, licensed under Academic Preprint / Open Access. Based on the abstract of the research paper. Full text available via ArXiv.