Nonconvex Stochastic Optimization under Heavy-Tailed Noises: Optimal Convergence without Gradient Clipping (2412.19529v1)

Abstract: Recently, the study of heavy-tailed noises in first-order nonconvex stochastic optimization has gotten a lot of attention since it was recognized as a more realistic condition as suggested by many empirical observations. Specifically, the stochastic noise (the difference between the stochastic and true gradient) is considered only to have a finite $\mathfrak{p}$-th moment where $\mathfrak{p}\in\left(1,2\right]$ instead of assuming it always satisfies the classical finite variance assumption. To deal with this more challenging setting, people have proposed different algorithms and proved them to converge at an optimal $\mathcal{O}(T^{\frac{1-\mathfrak{p}}{3\mathfrak{p}-2}})$ rate for smooth objectives after $T$ iterations. Notably, all these new-designed algorithms are based on the same technique - gradient clipping. Naturally, one may want to know whether the clipping method is a necessary ingredient and the only way to guarantee convergence under heavy-tailed noises. In this work, by revisiting the existing Batched Normalized Stochastic Gradient Descent with Momentum (Batched NSGDM) algorithm, we provide the first convergence result under heavy-tailed noises but without gradient clipping. Concretely, we prove that Batched NSGDM can achieve the optimal $\mathcal{O}(T^{\frac{1-\mathfrak{p}}{3\mathfrak{p}-2}})$ rate even under the relaxed smooth condition. More interestingly, we also establish the first $\mathcal{O}(T^{\frac{1-\mathfrak{p}}{2\mathfrak{p}}})$ convergence rate in the case where the tail index $\mathfrak{p}$ is unknown in advance, which is arguably the common scenario in practice.

• The paper introduces Batched NSGDM, which achieves optimal convergence without gradient clipping under heavy-tailed noise.
• It attains a convergence rate of O(T^((1-p)/(3p-2))) and robustly handles unknown tail indices with a rate of O(T^((1-p)/(2p))).
• The work extends classical assumptions by incorporating a generalized smoothness condition and a novel heavy-tailed noise model with vector-valued martingale inequalities.

Nonconvex Stochastic Optimization under Heavy-Tailed Noises: Optimal Convergence without Gradient Clipping

The paper presented by Zijian Liu and Zhengyuan Zhou explores the challenges associated with nonconvex stochastic optimization when confronted with heavy-tailed noise distributions. Traditionally, stochastic optimization methods have relied on the assumption of finite variance in noise. However, recent empirical studies suggest that these assumptions may be overly optimistic for modern machine learning tasks, where heavier-tailed noise is more prevalent. Confronted with this scenario, previous solutions have centered around gradient clipping techniques to ensure convergence. This work, however, explores the possibility of achieving convergence without resorting to gradient clipping by employing the Batched Normalized Stochastic Gradient Descent with Momentum (Batched NSGDM) algorithm.

Key Contributions

1. Convergence without Gradient Clipping: The core contribution of the paper is demonstrating that the Batched NSGDM algorithm can achieve optimal convergence rates under heavy-tailed noise conditions without any gradient clipping. The authors show that Batched NSGDM can achieve an $\mathcal{O}(T^{(1-p)/(3p-2)})$ convergence rate, where $T$ represents the number of iterations, and $p$ denotes the order of the moment. Notably, this convergence rate aligns with the lower bound established for optimal algorithms under heavy-tailed conditions.
2. Handling Unknown Tail Index $p$: The authors also address the practical scenario where the tail index $p$ is not known a priori. Here, they provide the first $\mathcal{O}(T^{(1-p)/(2p)})$ convergence rate, which demonstrates the robustness of Batched NSGDM to ambiguity in the tail index.
3. Generalized Smoothing and Noise Assumptions: The paper extends the classical smoothness and noise variance assumptions. The authors incorporate a generalized smoothness condition and present a new heavy-tailed noise assumption that is less restrictive and covers cases where traditional assumptions might fail.
4. Theoretical Insights: From a theoretical standpoint, the authors offer a novel expected inequality for vector-valued martingale difference sequences. This approach could be of independent interest for deriving high-probability bounds in stochastic optimization settings.

Implications and Future Directions

The findings from this paper bear significant implications. On a theoretical level, the results highlight the potential of gradient normalization techniques as an alternative to gradient clipping. Practically, this means that off-the-shelf stochastic gradient methods might be adapted with normalization to handle heavy-tailed noises efficiently—bypassing the need to calibrate clipping thresholds.

The paper opens avenues for future research. One exciting direction would be to explore whether the minimax rates for unknown tail indices can be improved further, possibly matching the known-index rates without prior knowledge. Another is the extension of similar techniques to adaptive gradient methods like Adam or AdaGrad, which are known for their robustness in practice but lack theoretical backing for heavy-tailed noise scenarios.

The theoretical framework presented in this paper for heavy-tailed noise management without clipping offers a valuable foundation for researchers seeking to build resilient optimization algorithms in the evolving landscape of machine learning.