Weak-$L^α$ Noise, Strong Guarantees: Gradient Clipping under Affine Heavy Tails
Gradient clipping is a key tool for obtaining high-probability convergence guarantees for stochastic first-order methods under heavy-tailed noise. Existing analyses typically assume a bounded central moment of order $α\in(1,2]$, which excludes distributions with tails of order $u^{-α}$ at the critical exponent. In this paper, we consider smooth convex optimization under affine heavy-tailed noise, whose scale may grow with the gradient norm. We derive high-probability convergence guarantees for clipped SGD (Clip-SGD) and its accelerated variant Clip-SSTM under a weak-$L^α$ tail condition, which is strictly weaker than the corresponding moment bound. For $1<α<2$, the clipping bounds have th...
| Authors | Alyona Nekrasova, Makar Sidorov, Savelii Chezhegov et al. |
|---|---|
| Primary category | math.OC |
| Published | 20 October 2026 |
| Last updated at source | 20 October 2026 |
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