Robust and Fast Training via Per-Sample Clipping

Gradient clipping is a standard safeguard for training neural networks under noisy, heavy-tailed stochastic gradients; yet, most clipping rules treat all parameters as vectors and ignore the matrix structure of modern architectures. We show empirically that data outliers often amplify only a small number of leading singular values in layer-wise gradient matrices, while the rest of the spectrum remains largely unchanged. Motivated by this phenomenon, we propose spectral clipping, which stabilizes training by clamping singular values that exceed a threshold while preserving the singular directions. This framework generalizes classical gradient norm clipping and can be easily integrated into existing optimizers. We provide a convergence analysis for non-convex optimization with spectrally clipped SGD, yielding the optimal $\mathcal{O}\left(K^{\frac{2 - 2α}{3α- 2}}\right)$ rate for heavy-tailed noise. To minimize hyperparameter tuning, we introduce layer-wise adaptive thresholds based on moving averages or sliding-window quantiles of the top singular values. Finally, we develop efficient implementations that clip only the top $r$ singular values via randomized truncated SVD, avoiding full decompositions for large layers. We demonstrate competitive performance across synthetic heavy-tailed settings and neural network training tasks.

In modern machine learning, parallelization of training is an important strategy for increasing scale. Asynchronous stochastic gradient descent (ASGD), which maximizes the utilization of available hardware by avoiding waiting for slow workers. However, with constant step sizes, the convergence of ASGD is nonetheless affected negatively by slow workers due to large delays in updates. At the same time, it has been empirically observed in asynchronous training of deep learning models that gradient clipping "stabilizes" training. In this work, we provide a theoretical justification for this behavior, as we show that clipping removes the dependence of the maximum delay in the oracle complexity. We employ a sub-Weibull model of gradient noise which generalizes sub-Gaussian and sub-exponential distributions to more heavy-tailed distributions, motivated by empirical observations in deep learning. We show convergence in expectation, and the first time in asynchronous optimization, convergence with high probability.

Classical convergence guarantees for stochastic gradient methods typically assume Lipschitz-smooth objectives and finite-variance gradient noise, both frequently violated in practice. In contrast, we study nonconvex stochastic optimization under the joint relaxation of these assumptions: objectives with $(L,s)$-H"older continuous gradients, $s\in(0,1]$, and gradient noise satisfying only a bounded $\alpha$-th moment condition for $\alpha\in(1,2]$. We establish three convergence results. Firstly, that standard SGD converges at rate $O(T^{-s/(1+s)})$ whenever $\alpha\ge1+s$, extending the classical nonconvex SGD rate to heavy-tailed noise and H"older smoothness simultaneously. Secondly, we analyze $\delta$-regularized gradient clipping ($\delta$-GClip), a provable trainer of wide and deep nets, and establish a stationarity rate of $O(T^{-2s(\alpha-1)/[(1+s)(2\alpha-1)]})$ under the same condition. Thirdly, we analyze standard gradient clipping (G-Clip) and show that it recovers the above rate for $\alpha\ge1+s$ while in the very heavy-tailed regime $\alpha<1+s$, it has a convergence rate $O(T^{-2s(\alpha-1)/[(\alpha-1)+s(2\alpha-1)]})$ --- the first convergence guarantee in this regime for any stochastic gradient based method.