# Algorithmic Stability of Heavy-Tailed SGD with General Loss Functions

arXiv: https://arxiv.org/abs/2301.11885 · HTML: https://mlsanity.com/paper/2301.11885/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 36 references that are in the map.
- First posted on arXiv 27 Jan 2023.

## Builds on (most cited first, 18 of 18)

- [Train faster, generalize better: Stability of stochastic gradient descent](https://mlsanity.com/paper/1509.01240/) (2015)
- [Information-theoretic analysis of generalization capability of learning algorithms](https://mlsanity.com/paper/1705.07809/) (2017)
- [Non-convex learning via Stochastic Gradient Langevin Dynamics: a nonasymptotic analysis](https://mlsanity.com/paper/1702.03849/) (2017)
- [Generalization Bounds of Stochastic Gradient Descent for Wide and Deep Neural Networks](https://mlsanity.com/paper/1905.13210/) (2019)
- [The Heavy-Tail Phenomenon in SGD](https://mlsanity.com/paper/2006.04740/) (2020)
- [Fine-Grained Analysis of Stability and Generalization for Stochastic Gradient Descent](https://mlsanity.com/paper/2006.08157/) (2020)
- [Information-Theoretic Generalization Bounds for Stochastic Gradient Descent](https://mlsanity.com/paper/2102.00931/) (2021)
- [Hausdorff Dimension, Heavy Tails, and Generalization in Neural Networks](https://mlsanity.com/paper/2006.09313/) (2020)
- [Global Non-convex Optimization with Discretized Diffusions](https://mlsanity.com/paper/1810.12361/) (2018)
- [First Exit Time Analysis of Stochastic Gradient Descent Under Heavy-Tailed Gradient Noise](https://mlsanity.com/paper/1906.09069/) (2019)
- [Heavy Tails in SGD and Compressibility of Overparametrized Neural Networks](https://mlsanity.com/paper/2106.03795/) (2021)
- [Convergence Rates of Stochastic Gradient Descent under Infinite Noise Variance](https://mlsanity.com/paper/2102.10346/) (2021)
- [Global Convergence of Stochastic Gradient Hamiltonian Monte Carlo for Non-Convex Stochastic Optimization: Non-Asymptotic Performance Bounds and Momentum-Based Acceleration](https://mlsanity.com/paper/1809.04618/) (2018)
- [Time-independent Generalization Bounds for SGLD in Non-convex Settings](https://mlsanity.com/paper/2111.12876/) (2021)
- [Improved Convergence Rate of Stochastic Gradient Langevin Dynamics with Variance Reduction and its Application to Optimization](https://mlsanity.com/paper/2203.16217/) (2022)
- [Algorithmic Stability of Heavy-Tailed Stochastic Gradient Descent on Least Squares](https://mlsanity.com/paper/2206.01274/) (2022)
- [Generalization Bounds for Stochastic Gradient Descent via Localized varepsilon-Covers](https://mlsanity.com/paper/2209.08951/) (2022)
- [Chaotic Regularization and Heavy-Tailed Limits for Deterministic Gradient Descent](https://mlsanity.com/paper/2205.11361/) (2022)

## Built on by (most cited first, 15 of 15)

- [Temperature Balancing, Layer-wise Weight Analysis, and Neural Network Training](https://mlsanity.com/paper/2312.00359/) (2023)
- [From Spikes to Heavy Tails: Unveiling the Spectral Evolution of Neural Networks](https://mlsanity.com/paper/2406.04657/) (2024)
- [Uniform-in-Time Wasserstein Stability Bounds for (Noisy) Stochastic Gradient Descent](https://mlsanity.com/paper/2305.12056/) (2023)
- [From Mutual Information to Expected Dynamics: New Generalization Bounds for Heavy-Tailed SGD](https://mlsanity.com/paper/2312.00427/) (2023)
- [Emergence of heavy tails in homogenized stochastic gradient descent](https://mlsanity.com/paper/2402.01382/) (2024)
- [Limit Theorems for Stochastic Gradient Descent with Infinite Variance](https://mlsanity.com/paper/2410.16340/) (2024)
- [Approximate Heavy Tails in Offline (Multi-Pass) Stochastic Gradient Descent](https://mlsanity.com/paper/2310.18455/) (2023)
- [Privacy of SGD under Gaussian or Heavy-Tailed Noise: Guarantees without Gradient Clipping](https://mlsanity.com/paper/2403.02051/) (2024)
- [Cyclic and Randomized Stepsizes Invoke Heavier Tails in SGD than Constant Stepsize](https://mlsanity.com/paper/2302.05516/) (2023)
- [Efficient Sampling of Stochastic Differential Equations with Positive Semi-Definite Models](https://mlsanity.com/paper/2303.17109/) (2023)
- [Learning Rate Should Scale Inversely with High-Order Data Moments in High-Dimensional Online Independent Component Analysis](https://mlsanity.com/paper/2509.15127/) (2025)
- [R'enyi Differential Privacy for Heavy-Tailed SDEs via Fractional Poincar'e Inequalities](https://mlsanity.com/paper/2511.15634/) (2025)
- [Stability and Generalization of Stochastic Nonconvex Optimization under Heavy-Tailed Noise: From Gradient Norm to Goldstein Stationarity](https://mlsanity.com/paper/2601.19730/) (2026)
- [Tail-Aware Information-Theoretic Bounds for LLM Alignment under Heavy-Tailed Rewards](https://mlsanity.com/paper/2604.10727/) (2026)
- [Dimension-Corrected Hitting Times for Heavy-Tailed Spectral Emergence in Neural Optimizer Dynamics](https://mlsanity.com/paper/2609.12994/) (2026)

## Abstract (arXiv)

Heavy-tail phenomena in stochastic gradient descent (SGD) have been reported in several empirical studies. Experimental evidence in previous works suggests a strong interplay between the heaviness of the tails and generalization behavior of SGD. To address this empirical phenomena theoretically, several works have made strong topological and statistical assumptions to link the generalization error to heavy tails. Very recently, new generalization bounds have been proven, indicating a non-monotonic relationship between the generalization error and heavy tails, which is more pertinent to the reported empirical observations. While these bounds do not require additional topological assumptions given that SGD can be modeled using a heavy-tailed stochastic differential equation (SDE), they can only apply to simple quadratic problems. In this paper, we build on this line of research and develop generalization bounds for a more general class of objective functions, which includes non-convex functions as well. Our approach is based on developing Wasserstein stability bounds for heavy-tailed SDEs and their discretizations, which we then convert to generalization bounds. Our results do not require any nontrivial assumptions; yet, they shed more light to the empirical observations, thanks to the generality of the loss functions.