In distributed stochastic optimization, many existing algorithms rely on the assumption that the objective function has a globally Lipschitz continuous gradient. However, this assumption may not hold in practical scenarios, particularly when the local smoothness constant may vary drastically across the domain, and sometimes even unbounded, using the global upper bound of the local constants is too conservative. To address this limitation, we propose a Relative Uniform Continuity (RUC) regularity condition for the local smoothness constant as a function of sets. The RUC condition covers most common growth functions for local smoothness constant, ranging from constant and logarithmic to polynomial and even exponential. For RUC-regular distributed optimization problems with finite-sum structure, we derive a clipped gradient tracking method with staggered variance reduction, which only relies on the local smoothness of objective functions. We also provide a convergence analysis demonstrating that the algorithm achieves a good convergence rate, offering an effective solution for decentralized stochastic optimization in non-Lipschitz smooth settings and highlighting their effectiveness in handling non-convex and non-Lipschitz smooth settings.

Mei Leilei is a PhD candidate in the Department of Industrial Systems Engineering and Management at the National University of Singapore, advised by Dr. Zhang Junyu. She received her Bachelor’s degree and Master’s degree from Southern University of Science and Technology. Her research focuses on distributed optimization, gradient tracking, non-Lipschitz-smooth problem.