# Large Deviation Upper Bounds and Improved MSE Rates of Nonlinear SGD: Heavy-tailed Noise and Power of Symmetry

arXiv: https://arxiv.org/abs/2410.15637 · HTML: https://mlsanity.com/paper/2410.15637/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 38 of its 79 references that are in the map.
- First posted on arXiv 21 Oct 2024.

## Builds on (most cited first, 30 of 38)

- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [Stochastic First- and Zeroth-order Methods for Nonconvex Stochastic Programming](https://mlsanity.com/paper/1309.5549/) (2013)
- [signSGD: Compressed Optimisation for Non-Convex Problems](https://mlsanity.com/paper/1802.04434/) (2018)
- [Why gradient clipping accelerates training: A theoretical justification for adaptivity](https://mlsanity.com/paper/1905.11881/) (2019)
- [Lower Bounds for Non-Convex Stochastic Optimization](https://mlsanity.com/paper/1912.02365/) (2019)
- [A Tail-Index Analysis of Stochastic Gradient Noise in Deep Neural Networks](https://mlsanity.com/paper/1901.06053/) (2019)
- [Better Theory for SGD in the Nonconvex World](https://mlsanity.com/paper/2002.03329/) (2020)
- [On the Convergence of Stochastic Gradient Descent with Adaptive Stepsizes](https://mlsanity.com/paper/1805.08114/) (2018)
- [Understanding Gradient Clipping in Private SGD: A Geometric Perspective](https://mlsanity.com/paper/2006.15429/) (2020)
- [Momentum Improves Normalized SGD](https://mlsanity.com/paper/2002.03305/) (2020)
- [The Heavy-Tail Phenomenon in SGD](https://mlsanity.com/paper/2006.04740/) (2020)
- [A Short Note on Concentration Inequalities for Random Vectors with SubGaussian Norm](https://mlsanity.com/paper/1902.03736/) (2019)
- [Tight Analyses for Non-Smooth Stochastic Gradient Descent](https://mlsanity.com/paper/1812.05217/) (2018)
- [Improved Analysis of Clipping Algorithms for Non-convex Optimization](https://mlsanity.com/paper/2010.02519/) (2020)
- [Robustness to Unbounded Smoothness of Generalized SignSGD](https://mlsanity.com/paper/2208.11195/) (2022)
- [On the Almost Sure Convergence of Stochastic Gradient Descent in Non-Convex Problems](https://mlsanity.com/paper/2006.11144/) (2020)
- [Beyond Convexity: Stochastic Quasi-Convex Optimization](https://mlsanity.com/paper/1507.02030/) (2015)
- [A High Probability Analysis of Adaptive SGD with Momentum](https://mlsanity.com/paper/2007.14294/) (2020)
- [High-Probability Bounds for Stochastic Optimization and Variational Inequalities: the Case of Unbounded Variance](https://mlsanity.com/paper/2302.00999/) (2023)
- [High Probability Convergence of Stochastic Gradient Methods](https://mlsanity.com/paper/2302.14843/) (2023)
- [Stochastic Gradient Descent for Nonconvex Learning without Bounded Gradient Assumptions](https://mlsanity.com/paper/1902.00908/) (2019)
- [Heavy Tails in SGD and Compressibility of Overparametrized Neural Networks](https://mlsanity.com/paper/2106.03795/) (2021)
- [On the Heavy-Tailed Theory of Stochastic Gradient Descent for Deep Neural Networks](https://mlsanity.com/paper/1912.00018/) (2019)
- [Revisiting the Noise Model of Stochastic Gradient Descent](https://mlsanity.com/paper/2303.02749/) (2023)
- [High Probability Convergence Bounds for Non-convex Stochastic Gradient Descent with Sub-Weibull Noise](https://mlsanity.com/paper/2006.05610/) (2020)
- [Breaking the Lower Bound with (Little) Structure: Acceleration in Non-Convex Stochastic Optimization with Heavy-Tailed Noise](https://mlsanity.com/paper/2302.06763/) (2023)
- [Breaking the Heavy-Tailed Noise Barrier in Stochastic Optimization Problems](https://mlsanity.com/paper/2311.04161/) (2023)
- [Stochastic Nonsmooth Convex Optimization with Heavy-Tailed Noises: High-Probability Bound, In-Expectation Rate and Initial Distance Adaptation](https://mlsanity.com/paper/2303.12277/) (2023)
- [What is the long-run distribution of stochastic gradient descent? A large deviations analysis](https://mlsanity.com/paper/2406.09241/) (2024)
- [Nonlinear gradient mappings and stochastic optimization: A general framework with applications to heavy-tail noise](https://mlsanity.com/paper/2204.02593/) (2022)

## Built on by (most cited first, 7 of 7)

- [Sharp High-Probability Rates for Nonlinear SGD under Heavy-Tailed Noise via Symmetrization](https://mlsanity.com/paper/2507.09093/) (2025)
- [Accelerated Gradient Methods with Biased Gradient Estimates: Risk Sensitivity, High-Probability Guarantees, and Large Deviation Bounds](https://mlsanity.com/paper/2509.13628/) (2025)
- [High-Probability Convergence Guarantees of Decentralized SGD](https://mlsanity.com/paper/2510.06141/) (2025)
- [Tight Long-Term Tail Decay of (Clipped) SGD in Non-Convex Optimization](https://mlsanity.com/paper/2602.05657/) (2026)
- [Bias-Variance Trade-off for Clipped Stochastic First-Order Methods: From Bounded Variance to Infinite Mean](https://mlsanity.com/paper/2512.14686/) (2025)
- [High-Probability Convergence in Decentralized Stochastic Optimization with Gradient Tracking](https://mlsanity.com/paper/2605.00281/) (2026)
- [A stochastic subgradient method with optimal failure exponent](https://mlsanity.com/paper/2609.37425/) (2026)

## Abstract (arXiv)

We study large deviation upper bounds and mean-squared error (MSE) guarantees of a general framework of nonlinear stochastic gradient methods in the online setting, in the presence of heavy-tailed noise. Unlike existing works that rely on the closed form of a nonlinearity (typically clipping), our framework treats the nonlinearity in a black-box manner, allowing us to provide unified guarantees for a broad class of bounded nonlinearities, including many popular ones, like sign, quantization, normalization, as well as component-wise and joint clipping. We provide several strong results for a broad range of step-sizes in the presence of heavy-tailed noise with symmetric probability density function, positive in a neighbourhood of zero and potentially unbounded moments. In particular, for non-convex costs we provide a large deviation upper bound for the minimum norm-squared of gradients, showing an asymptotic tail decay on an exponential scale, at a rate √t / log(t). We establish the accompanying rate function, showing an explicit dependence on the choice of step-size, nonlinearity, noise and problem parameters. Next, for non-convex costs and the minimum norm-squared of gradients, we derive the optimal MSE rate widetildeO(t-1/2). Moreover, for strongly convex costs and the last iterate, we provide an MSE rate that can be made arbitrarily close to the optimal rate O(t-1), improving on the state-of-the-art results in the presence of heavy-tailed noise. Finally, we establish almost sure convergence of the minimum norm-squared of gradients, providing an explicit rate, which can be made arbitrarily close to o(t-1/4).