Order-Optimal Projection-Free Algorithm for Adversarially Constrained Online Convex Optimization

2025-02-23

AI Summary

Projection-based methods for constrained online convex optimization (COCO) in high-dimensional adversarial settings suffer from poor scalability due to expensive projection computations. Method: We propose the first projection-free algorithm for COCO, integrating a separation oracle, adaptive online gradient descent, and a Lyapunov-type surrogate function; it jointly optimizes regret and cumulative constraint violation via dynamic step sizes and block-wise updates. Contributions/Results: (i) First projection-free guarantees of $O(sqrt{T})$ regret and $O(sqrt{T}log T)$ cumulative constraint violation for general convex losses, or $O(log T)$ regret and $O(sqrt{Tlog T})$ violation for strongly convex losses; (ii) explicit trade-off between oracle query complexity ($ ilde{O}(T)$) and performance; (iii) closure of the theoretical gap between projection-free and projection-based approaches.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Constraint Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Projection-based algorithms for constrained Online Convex Optimization (COCO) face scalability challenges in high-dimensional settings due to the computational complexity of projecting iterates onto constraint sets. This paper introduces a projection-free algorithm for COCO that achieves state-of-the-art performance guarantees while eliminating the need for projections. By integrating a separation oracle with adaptive Online Gradient Descent (OGD) and employing a Lyapunov-driven surrogate function, while dynamically adjusting step sizes using gradient norms, our method jointly optimizes the regret and cumulative constraint violation (CCV). We also use a blocked version of OGD that helps achieve tradeoffs betweeen the regret and CCV with the number of calls to the separation oracle. For convex cost functions, our algorithm attains an optimal regret of $mathcal{O}(sqrt{T})$ and a CCV of $mathcal{O}(sqrt{T} log T)$, matching the best-known projection-based results, while only using $ ilde{mathcal{O}}({T})$ calls to the separation oracle. The results also demonstrate a tradeoff where lower calls to the separation oracle increase the regret and the CCV. In the strongly convex setting, we further achieve a regret of $mathcal{O}(log T)$ and a CCV of $mathcal{O}(sqrt{Tlog T} )$, while requiring ${mathcal{O}}({T}^2)$ calls to the separation oracle. Further, tradeoff with the decreasing oracle calls is studied. These results close the gap between projection-free and projection-based approaches, demonstrating that projection-free methods can achieve performance comparable to projection-based counterparts.

Problem

• Develops projection-free COCO algorithm
• Optimizes regret and constraint violation
• Reduces calls to separation oracle

Innovation

• Projection-free algorithm for COCO
• Integration of separation oracle with OGD
• Lyapunov-driven surrogate function optimization