Nonconvex Decentralized Stochastic Bilevel Optimization under Heavy-Tailed Noises

2025-09-18

AI Summary

This work addresses the practical challenges in decentralized stochastic bilevel optimization where the lower-level objective is non-strongly convex and stochastic gradients exhibit heavy-tailed noise—potentially with infinite variance. We propose NR-DSBO, the first normalized stochastic variance-reduced algorithm for decentralized bilevel optimization that operates without gradient clipping. Our method integrates normalized gradient estimation with a decentralized variance reduction mechanism, enabling rigorous convergence guarantees for nonconvex bilevel problems without assuming strong convexity or finite gradient variance. Theoretically, NR-DSBO achieves a sublinear convergence rate under heavy-tailed noise. Empirically, it significantly outperforms existing methods in both communication efficiency and robustness to gradient corruption.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• 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

Abstract

Existing decentralized stochastic optimization methods assume the lower-level loss function is strongly convex and the stochastic gradient noise has finite variance. These strong assumptions typically are not satisfied in real-world machine learning models. To address these limitations, we develop a novel decentralized stochastic bilevel optimization algorithm for the nonconvex bilevel optimization problem under heavy-tailed noises. Specifically, we develop a normalized stochastic variance-reduced bilevel gradient descent algorithm, which does not rely on any clipping operation. Moreover, we establish its convergence rate by innovatively bounding interdependent gradient sequences under heavy-tailed noises for nonconvex decentralized bilevel optimization problems. As far as we know, this is the first decentralized bilevel optimization algorithm with rigorous theoretical guarantees under heavy-tailed noises. The extensive experimental results confirm the effectiveness of our algorithm in handling heavy-tailed noises.

Problem

• Addresses nonconvex decentralized bilevel optimization under heavy-tailed noises
• Develops normalized variance-reduced algorithm without clipping operations
• Establishes convergence guarantees for interdependent gradient sequences

Innovation

• Normalized stochastic variance-reduced gradient descent
• No clipping operation required
• Bounding interdependent gradient sequences theoretically