Gradient Clipping Beyond Vector Norms: A Spectral Approach for Matrix-Valued Parameters

• URL: http://arxiv.org/abs/2605.11838v1
• Date: Tue, 12 May 2026 09:24:59 GMT
• Title: Gradient Clipping Beyond Vector Norms: A Spectral Approach for Matrix-Valued Parameters
• Abstract summary: Most gradient clipping rules treat all parameters as heavy vectors and ignore the matrix structure of modern vectors.<n>We show empirically that data outliers often amplify only a small number of leading singular values.<n>Motivated by this phenomenon, we propose clipping, which stabilizes by singular values.
• Score: 48.3175117923623
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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