An Adaptive Decentralized Quasi-Newton Method with Stepsizes Independent of the Local-Update Budget
This paper proposes a novel Adaptive Decentralized Quasi-Newton (AdaDQN) method for solving smooth nonconvex optimization problems over undirected networks. While standard decentralized algorithms with multiple fixed local updates typically admit a convergence stepsize inversely proportional to the number of local updates, we show that this scaling is worst-case tight for the typical unscaled fixed-local-update scheme. We revisit a class of gradient-tracking methods with this scheme from a surrogate-based perspective and establish a Robust Inexact Algorithm (RIA) framework. Inspired by this framework, AdaDQN integrates a safeguarded consensus-aware termination criterion, a standard event-triggered communication protocol, and a scalable memoryless BFGS update. We establish an $\mathcal{O}(1/T)$ best-iterate rate for first-order stationarity. For a squared stationarity tolerance $\delta$, the guaranteed gradient complexity of the fixed scheme is $\mathcal{O}(nK_g/\delta)$, whereas AdaDQN attains $\mathcal{O}(n/\delta+n\alpha\tilde\varepsilon^{-2})$, independent of the maximum local-update budget ($K_g$). Numerical experiments demonstrate that AdaDQN achieves a superior computation-communication tradeoff, outperforming state-of-the-art decentralized methods across various performance metrics.