Achieving Near-Optimal Oracle Complexity in Decentralized Stochastic Optimization With Channel Noise

We study a decentralized nonconvex stochastic optimization problem in which a group of agents/nodes seek to minimize a global objective function that can be expressed as a sum of local component functions, such that each node in the network has access to exactly one component function. The nodes collaborate with their neighbors by sharing their estimates over communication links that are assumed to be corrupted by additive noise. To address this problem, we propose a computationally efficient and robust algorithm that builds on a probabilistic technique for stochastic gradient computation, which we believe will be applicable to a wide range of problems in decentralized information processing, learning, and control. Specifically, we show that the proposed method achieves an oracle complexity (computational complexity) of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$O(1/\epsilon ^{2})$</tex-math></inline-formula> for smooth and nonconvex functions with stochastic gradients, which is known to be sharp for its respective function class, and is an improvement over the computational cost obtained in previous works. In addition, we retain the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$O(1/\epsilon ^{3})$</tex-math></inline-formula> rate for the communication cost, which is at par with the communication cost obtained in previous works. We also show how the proposed algorithm has robust performance in environments with unreliable computational resources. Finally, the theoretical findings are validated via numerical experiments.