Achieving Better Local Regret Bound for Online Non-Convex Bilevel Optimization

2026-02-06

AI Summary

This work addresses the suboptimality and unclear optimality of existing local regret bounds in online nonconvex bilevel optimization. It establishes, for the first time, tight lower bounds for both standard and window-averaged local regret—specifically, Ω(1 + V_T) and Ω(T/W²), respectively—thereby characterizing the fundamental limits of performance in this setting. To match these bounds, the paper proposes an efficient single-loop algorithm that improves inner-level gradient query efficiency and incorporates a sliding window mechanism to adapt to environmental changes. The method achieves theoretical optimality while significantly reducing gradient evaluation complexity. Empirical results demonstrate the algorithm’s superiority in both theoretical guarantees and practical performance.

Technology Category

• Search and Optimization: Local Search
• Machine Learning: Online Learning & Bandits
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Online bilevel optimization (OBO) has emerged as a powerful framework for many machine learning problems. Prior works have developed several algorithms that minimize the standard bilevel local regret or the window-averaged bilevel local regret of the OBO problem, but the optimality of existing regret bounds remains unclear. In this work, we establish optimal regret bounds for both settings. For standard bilevel local regret, we propose an algorithm that achieves the optimal regret $\Omega(1+V_T)$ with at most $O(T\log T)$ total inner-level gradient evaluations. We further develop a fully single-loop algorithm whose regret bound includes an additional gradient-variation terms. For the window-averaged bilevel local regret, we design an algorithm that captures sublinear environmental variation through a window-based analysis and achieves the optimal regret $\Omega(T/W^2)$. Experiments validate our theoretical findings and demonstrate the practical effectiveness of the proposed methods.

Problem

• online bilevel optimization
• local regret
• non-convex optimization
• regret bound
• gradient evaluation

Innovation

• online bilevel optimization
• optimal regret bound
• non-convex optimization
• single-loop algorithm
• window-averaged regret