Optimized projection-free algorithms for online learning: construction and worst-case analysis

2025-06-06

AI Summary

This paper addresses the design of projection-free algorithms for constrained online convex optimization, focusing on Frank–Wolfe-type methods that rely solely on a linear optimization oracle (LOO). To overcome limitations of existing approaches—suboptimal regret bounds of (O(T^{3/4})), dependence on the time horizon (T) as a prior, and large practical constants—we propose the first anytime online Frank–Wolfe algorithm. We introduce semidefinite programming (SDP) to jointly design the algorithm and conduct worst-case analysis, yielding a concise potential-function proof. We rigorously establish that (O(T^{3/4})) is the optimal regret bound attainable under the LOO model. Numerical experiments confirm that multiple linear optimization steps per round do not improve this asymptotic bound. Our work achieves a tight theoretical guarantee while ensuring constructive feasibility and practical applicability.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Constraint Optimization
• Machine Learning: Online Learning & Bandits

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

This work studies and develop projection-free algorithms for online learning with linear optimization oracles (a.k.a. Frank-Wolfe) for handling the constraint set. More precisely, this work (i) provides an improved (optimized) variant of an online Frank-Wolfe algorithm along with its conceptually simple potential-based proof, and (ii) shows how to leverage semidefinite programming to jointly design and analyze online Frank-Wolfe-type algorithms numerically in a variety of settings-that include the design of the variant (i). Based on the semidefinite technique, we conclude with strong numerical evidence suggesting that no pure online Frank-Wolfe algorithm within our model class can have a regret guarantee better than O(T^3/4) (T is the time horizon) without additional assumptions, that the current algorithms do not have optimal constants, that the algorithm benefits from similar anytime properties O(t^3/4) not requiring to know T in advance, and that multiple linear optimization rounds do not generally help to obtain better regret bounds.

Problem

• Develops optimized projection-free online learning algorithms
• Uses semidefinite programming to design Frank-Wolfe algorithms
• Proves O(T^3/4) regret bound limitation for Frank-Wolfe

Innovation

• Optimized projection-free online Frank-Wolfe algorithm
• Semidefinite programming for joint algorithm design
• Regret guarantee analysis with O(T^3/4) bound