Tight Lower Bounds and Optimal Algorithms for Stochastic Nonconvex Optimization with Heavy-Tailed Noise

Abstract: We study stochastic nonconvex optimization under heavy-tailed noise. In this setting, the stochastic gradients only have bounded $p$--th central moment ($p$--BCM) for some $p \in (1,2]$. Building on the foundational work of Arjevani et al. (2022) in stochastic optimization, we establish tight sample complexity lower bounds for all first-order methods under \emph{relaxed} mean-squared smoothness ($q$-WAS) and $δ$-similarity ($(q, δ)$-S) assumptions, allowing any exponent $q \in [1,2]$ instead of the standard $q = 2$. These results substantially broaden the scope of existing lower bounds. To complement them, we show that Normalized Stochastic Gradient Descent with Momentum Variance Reduction (NSGD-MVR), a known algorithm, matches these bounds in expectation. Beyond expectation guarantees, we introduce a new algorithm, Double-Clipped NSGD-MVR, which allows the derivation of high-probability convergence rates under weaker assumptions than previous works. Finally, for second-order methods with stochastic Hessians satisfying bounded $q$-th central moment assumptions for some exponent $q \in [1, 2]$ (allowing $q \neq p$), we establish sharper lower bounds than previous works while improving over Sadiev et al. (2025) (where only $p = q$ is considered) and yielding stronger convergence exponents. Together, these results provide a nearly complete complexity characterization of stochastic nonconvex optimization in heavy-tailed regimes.