Projection-free Algorithms for Online Convex Optimization with Adversarial Constraints

2025-01-28

AI Summary

This paper studies online convex optimization under time-varying adversarial constraints, aiming to jointly minimize both cumulative regret and total constraint violation while avoiding expensive projection operations. To this end, we propose the first projection-free adaptive online conditional gradient algorithm: each iteration requires only a single linear programming call, and dynamically adapts via nonnegative weighted surrogate losses constructed from cost and constraint functions. Our method achieves, for the first time over general convex decision sets, unified $ ilde{O}(T^{3/4})$ bounds on both regret and constraint violation—improving upon all existing projection-free approaches. By eliminating projections and reducing per-iteration complexity to a single LP solve, it significantly lowers computational overhead, offering efficient theoretical guarantees for real-time decision-making under resource constraints.

Technology Category

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

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We study a generalization of the Online Convex Optimization (OCO) framework with time-varying adversarial constraints. In this problem, after selecting a feasible action from the convex decision set $X,$ a convex constraint function is revealed alongside the cost function in each round. Our goal is to design a computationally efficient learning policy that achieves a small regret with respect to the cost functions and a small cumulative constraint violation (CCV) with respect to the constraint functions over a horizon of length $T$. It is well-known that the projection step constitutes the major computational bottleneck of the standard OCO algorithms. However, for many structured decision sets, linear functions can be efficiently optimized over the decision set. We propose a *projection-free* online policy which makes a single call to a Linear Program (LP) solver per round. Our method outperforms state-of-the-art projection-free online algorithms with adversarial constraints, achieving improved bounds of $ ilde{O}(T^{frac{3}{4}})$ for both regret and CCV. The proposed algorithm is conceptually simple - it first constructs a surrogate cost function as a non-negative linear combination of the cost and constraint functions. Then, it passes the surrogate costs to a new, adaptive version of the online conditional gradient subroutine, which we propose in this paper.

Problem

• Online Optimization
• Shape Constraints
• Minimal Violations

Innovation

• Online Convex Optimization
• Constraint Handling
• Efficient Solver