Finding Local Minima Efficiently in Decentralized Optimization

In this paper we study the second-order optimality of decentralized stochastic algorithm that escapes saddle point efficiently for nonconvex optimization problems. We propose a new pure gradient-based decentralized stochastic algorithm PEDESTAL with a novel convergence analysis framework to address the technical challenges unique to the decentralized stochastic setting. Our method is the first decentralized stochastic algorithm to achieve second-order optimality with non-asymptotic analysis. We provide theoretical guarantees with the gradient complexity of $\tilde{O} (\epsilon^{-3})$ to find $O(\epsilon, \sqrt{\epsilon})$-second-order stationary point, which matches state-of-the-art results of centralized counterparts or decentralized methods to find first-order stationary point. We also conduct two decentralized tasks in our experiments, a matrix sensing task with synthetic data and a matrix factorization task with a real-world dataset to validate the performance of our method.