Complexity of Normalized Stochastic First-Order Methods with Momentum Under Heavy-Tailed Noise

In this paper, we propose practical normalized stochastic first-order methods with Polyak momentum, multiextrapolated momentum, and recursive momentum for solving unconstrained optimization problems. These methods employ dynamically updated algorithmic parameters and do not require explicit knowledge of problem-dependent quantities such as the Lipschitz constant or noise bound. We establish first-order oracle complexity results for finding approximate stochastic stationary points under heavy-tailed noise and weakly average smoothness conditions—both of which are weaker than the commonly used bounded variance and mean-squared smoothness assumptions. Our complexity bounds either improve upon or match the best-known results in the literature. Numerical experiments are presented to demonstrate the practical effectiveness of the proposed methods. Funding: C. He was partially supported by the Wallenberg AI, Autonomous Systems and Software Program funded by the Knut and Alice Wallenberg Foundation. D. Sun was partially supported by the Research Center for Intelligent Operations Research at The Hong Kong Polytechnic University [Grant P0051214].