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

Abstract

We study the last iterate of clipped SGD, and of every rule that queries its current iterate and steps along the resulting stochastic gradient, when training is stopped at a data-dependent time, so that guarantees must hold at all times simultaneously. On smooth convex objectives under heavy-tailed noise with a bounded central $p$th moment, $1<p\le 2$, we determine the optimal time-uniform rate of these rules. A clipped SGD schedule without horizon or confidence inputs attains $\tilde{O}(t^{-{\beta }_p})$ at all times, with ${\beta }_p=2(p-1)/(3p-1)$, also almost surely, and no such rule attains $\tilde{O}(t^{-r})$ for any $r>{\beta }_p$, even with adaptive step sizes and thresholds. This rate lies between the optimal rate $O(T^{-{\alpha }_p})$ for a known horizon $T$, ${\alpha }_p=(p-1)/p$, and the previous time-uniform rate $\tilde{O}(t^{-2(p-1)/(3p)})$ (Chezhegov et al., ICLR 2026); under finite variance, the exponents are $1/2$, $2/5$, and $1/3$. Their conjecture that a better schedule reaches the optimal rate thus holds with a known horizon (Liu; Jiang et al.) but fails without one. Querying gradients at the running average attains $\tilde{O}(t^{-{\alpha }_p})$ at all times, whereas heavy-ball momentum with geometric forgetting and polynomially bounded steps cannot improve on ${\beta }_p$.

arXiv comment44 pages, including appendices and references. Lean 4 proofs of Theorems 2, 4, and 8 and of the known-horizon bound of Theorem 1, and the script of the numerical illustration, are included as ancillary files; the Lean code is also at https://github.com/Thefootofhappiness/clipped-sgd-last-iterate-lean