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

We propose a robust gradient estimator based on per-sample gradient clipping and analyze its properties both theoretically and empirically. We show that the resulting method, per-sample clipped SGD (PS-Clip-SGD), achieves optimal in-expectation convergence rates for non-convex optimization problems under heavy-tailed gradient noise. Moreover, we establish high-probability convergence guarantees that match the in-expectation rates up to polylogarithmic factors in the failure probability. We complement our theoretical results with multiple numerical experiments. In particular, we demonstrate that PS-Clip-SGD outperforms both vanilla SGD with momentum and standard gradient clipping when training AlexNet on the CIFAR-100 dataset, even after accounting for the additional computational time caused by per-sample clipping. We also empirically show that, in the presence of gradient accumulation, applying clipping at the mini-batch level can improve training performance while incurring virtually no additional computational cost. This finding is particularly interesting, as it contradicts the common practice of applying clipping only after all accumulation steps have been completed.

Stochastic gradient descent (SGD) with gradient clipping and additive noise has become a standard technique for training machine learning models, particularly in applications requiring robustness or privacy guarantees. However, clipping introduces a bias in stochastic gradients, while additive noise introduces additional variance, making the long-run behaviour of individual optimization trajectories difficult to characterize. In this work, we prove that SGD with clipping and additive Gaussian noise (SGD-CN) converges almost surely (a.s.) under smoothness and uniformly bounded stochastic-gradient noise assumptions, provided the step sizes satisfy some standard decaying conditions. Our analysis extends to momentum variants such as the stochastic heavy ball and Nesterov's accelerated gradient, where we show that careful energy constructions yield similar guarantees. These results provide stronger theoretical foundations for understanding the pathwise behaviour of clipped stochastic gradient methods and suggest that, despite the bias and noise introduced by clipping and perturbation, the algorithm remains stable in both convex and nonconvex regimes.

Stochastic gradient descent (SGD) is widely used to train machine learning models, but subsampling the training data introduces noise into its updates. The strength and applicability of high-probability guarantees therefore depend critically on how the tails of gradient noise are modeled. Reports of heavy-tailed gradient noise in deep learning motivate relaxing the bounded-noise and sub-Gaussian assumptions commonly used in high-probability analyses of SGD. We use Young functions from Orlicz space theory to describe noise tails in a common framework. We model SGD gradient noise by adopting a Young function that preserves the finiteness of all polynomial moments while allowing tails heavier than sub-Weibull, including lognormal distributions. The resulting class is called $β$-heavy-tailed, with $β$ controlling the tail heaviness. We establish concentration inequalities for $β$-heavy-tailed noise and combine them with a uniform bound on the difference between empirical and population gradients along the SGD trajectory to obtain high-probability bounds on optimization and population-risk stationarity for smooth nonconvex losses under trajectory assumptions. The bounds are not restricted to a particular learning-rate decay rule and make explicit the effects of noise tails and learning-rate schedules. Under the Polyak-Łojasiewicz condition, we bound the risk at the last iterate. We also analyze SGD with gradient clipping under the $β$-heavy-tailed noise model.