# Clipped or Unclipped? Finite-Sample Trade-offs for Averaged SGD under Heavy-Tailed Noise

arXiv: https://arxiv.org/abs/2609.32962 · HTML: https://mlsanity.com/paper/2609.32962/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 22 of its 33 references that are in the map.
- First posted on arXiv 26 Sep 2026.

## Builds on (most cited first, 22 of 22)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Llama 2: Open Foundation and Fine-Tuned Chat Models](https://mlsanity.com/paper/2307.09288/) (2023)
- [Stochastic Optimization with Heavy-Tailed Noise via Accelerated Gradient Clipping](https://mlsanity.com/paper/2005.10785/) (2020)
- [Overparameterized Nonlinear Learning: Gradient Descent Takes the Shortest Path?](https://mlsanity.com/paper/1812.10004/) (2018)
- [Revisiting Gradient Clipping: Stochastic bias and tight convergence guarantees](https://mlsanity.com/paper/2305.01588/) (2023)
- [Open Bandit Dataset and Pipeline: Towards Realistic and Reproducible Off-Policy Evaluation](https://mlsanity.com/paper/2008.07146/) (2020)
- [On Linear Stochastic Approximation: Fine-grained Polyak-Ruppert and Non-Asymptotic Concentration](https://mlsanity.com/paper/2004.04719/) (2020)
- [High-Probability Bounds for Stochastic Optimization and Variational Inequalities: the Case of Unbounded Variance](https://mlsanity.com/paper/2302.00999/) (2023)
- [Sub-Gaussian estimators of the mean of a random vector](https://mlsanity.com/paper/1702.00482/) (2017)
- [Finite-time High-probability Bounds for Polyak-Ruppert Averaged Iterates of Linear Stochastic Approximation](https://mlsanity.com/paper/2207.04475/) (2022)
- [Bias and Extrapolation in Markovian Linear Stochastic Approximation with Constant Stepsizes](https://mlsanity.com/paper/2210.00953/) (2022)
- [Tight High Probability Bounds for Linear Stochastic Approximation with Fixed Stepsize](https://mlsanity.com/paper/2106.01257/) (2021)
- [High-Probability Convergence for Composite and Distributed Stochastic Minimization and Variational Inequalities with Heavy-Tailed Noise](https://mlsanity.com/paper/2310.01860/) (2023)
- [High Probability Convergence of Clipped-SGD Under Heavy-tailed Noise](https://mlsanity.com/paper/2302.05437/) (2023)
- [Improved High-Probability Bounds for the Temporal Difference Learning Algorithm via Exponential Stability](https://mlsanity.com/paper/2310.14286/) (2023)
- [Gaussian Approximation and Multiplier Bootstrap for Polyak-Ruppert Averaged Linear Stochastic Approximation with Applications to TD Learning](https://mlsanity.com/paper/2405.16644/) (2024)
- [Can SGD Handle Heavy-Tailed Noise?](https://mlsanity.com/paper/2508.04860/) (2025)
- [Statistical inference for Linear Stochastic Approximation with Markovian Noise](https://mlsanity.com/paper/2505.19102/) (2025)
- [General Tail Bounds for Non-Smooth Stochastic Mirror Descent](https://mlsanity.com/paper/2312.07142/) (2023)
- [To Clip or not to Clip: the Dynamics of SGD with Gradient Clipping in High-Dimensions](https://mlsanity.com/paper/2406.11733/) (2024)
- [Accelerated stochastic first-order method for convex optimization under heavy-tailed noise](https://mlsanity.com/paper/2510.11676/) (2025)
- [In-Expectation Convergence of Stochastic Gradient Methods under Heavy-Tailed Noise](https://mlsanity.com/paper/2606.00520/) (2026)

## Abstract (arXiv)

Gradient clipping is widely used to stabilize training, but it need not improve the statistical accuracy of averaged SGD, even under heavy-tailed noise. We derive a finite-sample comparison of clipped and unclipped Polyak-Ruppert averaged SGD under finite conditional p-th moments, p≥2. Our main result gives explicit accuracy and confidence conditions under which, for p>2, the Gaussian term dominates the unclipped deviation bound, so clipping need not improve its leading order. By balancing clipping bias and concentration, we obtain a bound in which the heavy-tail correction depends logarithmically rather than polynomially on the inverse failure probability. At p=2, this improves the confidence dependence of the leading bound. We establish sharpness of the unclipped heavy-tail term through an exact one-dimensional quadratic recursion and extend the comparison to projected convex SGD. We also prove concrete costs of clipping: every fixed finite threshold increases asymptotic variance on a scalar Gaussian quadratic, while whole-gradient clipping can shift the limiting point under asymmetric noise.