On the Communication Complexity of Decentralized Bilevel Optimization

TL;DR

This paper introduces two new decentralized stochastic bilevel optimization algorithms that achieve faster convergence and lower communication costs in heterogeneous settings, with theoretical guarantees and empirical validation.

Contribution

The paper presents the first decentralized bilevel algorithms with convergence guarantees under mild heterogeneity assumptions, improving efficiency over existing methods.

Findings

Faster convergence rates than existing methods.

Lower communication costs in heterogeneous environments.

Empirical results confirm algorithm efficacy.

Abstract

Stochastic bilevel optimization finds widespread applications in machine learning, including meta-learning, hyperparameter optimization, and neural architecture search. To extend stochastic bilevel optimization to distributed data, several decentralized stochastic bilevel optimization algorithms have been developed. However, existing methods often suffer from slow convergence rates and high communication costs in heterogeneous settings, limiting their applicability to real-world tasks. To address these issues, we propose two novel decentralized stochastic bilevel gradient descent algorithms based on simultaneous and alternating update strategies. Our algorithms can achieve faster convergence rates and lower communication costs than existing methods. Importantly, our convergence analyses do not rely on strong assumptions regarding heterogeneity. More importantly, our theoretical analysis clearly discloses how the additional communication required for estimating hypergradient under the heterogeneous setting affects the convergence rate. To the best of our knowledge, this is the first time such favorable theoretical results have been achieved with mild assumptions in the heterogeneous setting. Furthermore, we demonstrate how to establish the convergence rate for the alternating update strategy when combined with the variance-reduced gradient. Finally, experimental results confirm the efficacy of our algorithms.