The Cost of an Unknown Horizon for the Last Iterate of Clipped SGD under Heavy-Tailed Noise

In this paper, we study the last-iterate convergence of clipped stochastic gradient descent under heavy-tailed noise. We assume that the stochastic gradients are unbiased and that the noise has a bounded central $p$-th moment, where $1<p\le2$. For smooth convex objectives, a linearly decreasing learning rate gives a high-probability error bound of order $O(LR^2\log(1/δ)/T+σR(\log(1/δ)/T)^{(p-1)/p})$. We further establish last-iterate guarantees for nonsmooth composite objectives and smooth composite objectives with proximal updates. In addition, when a causal adversary replaces at most $q$ gradients, the same clipped update achieves a bound with $\log(1/δ)$ replaced by $\log(1/δ)+q$. The corresponding lower bounds for the clean and corrupted oracle models are also provided.

In stochastic (strongly) convex optimization, establishing last-iterate convergence guarantees for first-order methods has recently garnered increasing attention, as this not only deepens the theoretical understanding of these algorithms but also aligns more closely with practice. Notably, people have demonstrated that one of the most famous, simple, and popular methods, Stochastic Gradient Descent ($\mathtt{SGD}$), provably achieves last-iterate convergence. However, the existing results may have limited applicability, since they typically rely on strong conditions on the gradient noise (e.g., an exponentially decaying tail) rather than the more realistic heavy-tailed noise. Unfortunately, when faced with heavy-tailed noise, $\mathtt{SGD}$ is known to exhibit undesirable behavior or even fail to converge. To address the challenge of heavy-tailed noise, people have proposed $\mathtt{Clipped}\text{-}\mathtt{SGD}$, an algorithm that combines $\mathtt{SGD}$ with a simple mechanism, gradient clipping. Although $\mathtt{Clipped}\text{-}\mathtt{SGD}$ performs well in practice, its last-iterate convergence property remains highly underexplored. In this work, we prove the first optimal last-iterate rates in high probability for $\mathtt{Clipped}\text{-}\mathtt{SGD}$ under heavy-tailed noise, thereby closing a gap in the literature.

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 the same powers of the clipping level as in the moment case, and both methods attain the additive stochastic rate $K^{-(α-1)/α}$. The gradient-dependent noise contributes a faster-decaying term of order $K^{-2(α-1)/α}$, up to logarithmic factors. At $α=2$, we characterize the additional logarithmic factor on the Lorentz scale and establish its necessity for the additive error at fixed confidence. We also analyze Clip-SGD with prescribed clipping levels and derive finite-horizon guarantees for reaching a neighborhood determined by the clipping bias. For additive noise, matching lower bounds show that this neighborhood is unavoidable, up to constants and the natural error cap. Numerical experiments with Pareto noise illustrate the theoretical results.