# Second-order Optimization under Heavy-Tailed Noise: Hessian Clipping and Sample Complexity Limits

arXiv: https://arxiv.org/abs/2510.10690 · HTML: https://mlsanity.com/paper/2510.10690/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 48 of its 64 references that are in the map.
- First posted on arXiv 12 Oct 2025.

## Builds on (most cited first, 30 of 48)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Deep Learning with Differential Privacy](https://mlsanity.com/paper/1607.00133/) (2016)
- [Lower Bounds for Non-Convex Stochastic Optimization](https://mlsanity.com/paper/1912.02365/) (2019)
- [Momentum-Based Variance Reduction in Non-Convex SGD](https://mlsanity.com/paper/1905.10018/) (2019)
- [A Tail-Index Analysis of Stochastic Gradient Noise in Deep Neural Networks](https://mlsanity.com/paper/1901.06053/) (2019)
- [SGD: General Analysis and Improved Rates](https://mlsanity.com/paper/1901.09401/) (2019)
- [Better Theory for SGD in the Nonconvex World](https://mlsanity.com/paper/2002.03329/) (2020)
- [Momentum Improves Normalized SGD](https://mlsanity.com/paper/2002.03305/) (2020)
- [Lower Bounds for Finding Stationary Points I](https://mlsanity.com/paper/1710.11606/) (2017)
- [Stochastic Optimization with Heavy-Tailed Noise via Accelerated Gradient Clipping](https://mlsanity.com/paper/2005.10785/) (2020)
- [Linear attention is (maybe) all you need (to understand transformer optimization)](https://mlsanity.com/paper/2310.01082/) (2023)
- [High-probability Bounds for Non-Convex Stochastic Optimization with Heavy Tails](https://mlsanity.com/paper/2106.14343/) (2021)
- [High-Probability Bounds for Stochastic Optimization and Variational Inequalities: the Case of Unbounded Variance](https://mlsanity.com/paper/2302.00999/) (2023)
- [Stochastic Policy Gradient Methods: Improved Sample Complexity for Fisher-non-degenerate Policies](https://mlsanity.com/paper/2302.01734/) (2023)
- [Stochastic Cubic Regularization for Fast Nonconvex Optimization](https://mlsanity.com/paper/1711.02838/) (2017)
- [High Probability Convergence of Stochastic Gradient Methods](https://mlsanity.com/paper/2302.14843/) (2023)
- [From Gradient Clipping to Normalization for Heavy Tailed SGD](https://mlsanity.com/paper/2410.13849/) (2024)
- [Stochastic Gradient Descent for Nonconvex Learning without Bounded Gradient Assumptions](https://mlsanity.com/paper/1902.00908/) (2019)
- [Nonconvex Stochastic Optimization under Heavy-Tailed Noises: Optimal Convergence without Gradient Clipping](https://mlsanity.com/paper/2412.19529/) (2024)
- [Second-Order Information in Non-Convex Stochastic Optimization: Power and Limitations](https://mlsanity.com/paper/2006.13476/) (2020)
- [Revisiting the Noise Model of Stochastic Gradient Descent](https://mlsanity.com/paper/2303.02749/) (2023)
- [Convergence Rates of Stochastic Gradient Descent under Infinite Noise Variance](https://mlsanity.com/paper/2102.10346/) (2021)
- [Algorithms of Robust Stochastic Optimization Based on Mirror Descent Method](https://mlsanity.com/paper/1907.02707/) (2019)
- [On Proximal Policy Optimization's Heavy-tailed Gradients](https://mlsanity.com/paper/2102.10264/) (2021)
- [Parameter-free Regret in High Probability with Heavy Tails](https://mlsanity.com/paper/2210.14355/) (2022)
- [Breaking the Lower Bound with (Little) Structure: Acceleration in Non-Convex Stochastic Optimization with Heavy-Tailed Noise](https://mlsanity.com/paper/2302.06763/) (2023)
- [Oracle Complexity of Second-Order Methods for Smooth Convex Optimization](https://mlsanity.com/paper/1705.07260/) (2017)
- [Stochastic Newton and Cubic Newton Methods with Simple Local Linear-Quadratic Rates](https://mlsanity.com/paper/1912.01597/) (2019)
- [High-Probability Convergence for Composite and Distributed Stochastic Minimization and Variational Inequalities with Heavy-Tailed Noise](https://mlsanity.com/paper/2310.01860/) (2023)
- [Two Sides of One Coin: the Limits of Untuned SGD and the Power of Adaptive Methods](https://mlsanity.com/paper/2305.12475/) (2023)

## Built on by (most cited first, 3 of 3)

- [Improved Convergence in Parameter-Agnostic Error Feedback through Momentum](https://mlsanity.com/paper/2511.14501/) (2025)
- [RanSOM: Second-Order Momentum with Randomized Scaling for Constrained and Unconstrained Optimization](https://mlsanity.com/paper/2602.06824/) (2026)
- [Bias-Variance Trade-off for Clipped Stochastic First-Order Methods: From Bounded Variance to Infinite Mean](https://mlsanity.com/paper/2512.14686/) (2025)

## Abstract (arXiv)

Heavy-tailed noise is pervasive in modern machine learning applications, arising from data heterogeneity, outliers, and non-stationary stochastic environments. While second-order methods can significantly accelerate convergence in light-tailed or bounded-noise settings, such algorithms are often brittle and lack guarantees under heavy-tailed noise -- precisely the regimes where robustness is most critical. In this work, we take a first step toward a theoretical understanding of second-order optimization under heavy-tailed noise. We consider a setting where stochastic gradients and Hessians have only bounded p-th moments, for some p∈ (1,2], and establish tight lower bounds on the sample complexity of any second-order method. We then develop a variant of normalized stochastic gradient descent that leverages second-order information and provably matches these lower bounds. To address the instability caused by large deviations, we introduce a novel algorithm based on gradient and Hessian clipping, and prove high-probability upper bounds that nearly match the fundamental limits. Our results provide the first comprehensive sample complexity characterization for second-order optimization under heavy-tailed noise. This positions Hessian clipping as a robust and theoretically sound strategy for second-order algorithm design in heavy-tailed regimes.