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 O(ϵ 3)to find O(ϵ, ϵ)-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.

In this paper we study the second-order optimality of decentralized stochastic algorithm that escapes saddle point efficiently for nonconvex optimization problems.

Today, India's Chandrayaan-3 became the first spacecraft to successfully land near the lunar south pole, and India became the fourth country to make a soft landing anywhere on lunar soil, following the former Soviet Union, the United States, and China. The robotic vehicle touched down at 8:33 Eastern time, nearly six weeks after its launch. The craft includes a four-legged lander and a small rover to study the lunar regolith and look for signs of water ice during a two-week mission. On August 20, the craft malfunctioned and appears to have crashed while preparing for a landing planned for the next day. Roscosmos, Russia's space agency, intended to deploy Luna-25 for a year-long mission near the Boguslavsky impact crater, where its eight scientific instruments would also have examined properties of the regolith and pockets of water ice.

The selected landing site should be at a low slope with respect to the planetary equipotential surface, free of hazards, and also be located such that it satisfies mission objectives. The Apollo Lunar missions allowed the lander pilot to steer towards a manually selected landing site by manipulating a control stick until the designated landing site (DLS) appeared in a window fixed reticle [1], but the actual trajectory flown to reach that point in a manner consistent with a soft landing was automated by the guidance and control system. The Chinese Chang'e 3 mission [2] used coarse hazard detection from greyscale images during the powered descent, and once the lander reached an altitude of 100 m, hovered while a hazard free landing site was selected, after which the lander diverted horizontally until directly above the selected landing site, and then continued straight down to the surface. Finally, the Morpheus project [3] demonstrated integration of flash LIDAR based hazard detection with guidance and control on Earth using a hazard field with a mix of terrain hazards and safe landing sites. During the powered descent the landing site detection system will likely change the DLS as distance to surface decreases and resolution increases. Consequently, a suitable guidance and control system should be capable of multiple divert maneuvers during the powered descent phase.