Why Clipping Matters in AdaGrad? Toward a High-Probability Theory under Generalized Smoothness

Title:Why Clipping Matters in AdaGrad? Toward a High-Probability Theory under Generalized Smoothness

Abstract:We analyze the original same-step coordinate-wise AdaGrad under generalized smoothness and heavy-tailed noise with bounded variance. In this setting, local curvature may grow sub-quadratically with the gradient norm, and stochastic gradients are assumed to have only bounded conditional second moments. We show that unclipped AdaGrad can become \emph{anisotropically miscalibrated}: under heavy-tailed noise, the adaptive denominator can learn the geometry of rare noise shocks rather than the local curvature of the objective, leading to a persistent directional distortion that blocks finite-horizon Euclidean progress. We then prove that clipping repairs this failure mode. Our main result is a finite-horizon high-probability guarantee for the original non-lagged AdaGrad update, yielding $\frac1T\sum_{t=0}^{T-1}\|\nabla f(x_t)\|^2=\mathcal{O}\left(\frac{d\big(\sqrt{\log T} + \log \frac{1}{\delta}\big)}{\sqrt{T}}\right),$ and hence $\widetilde{\mathcal O}(\varepsilon^{-2})$ complexity. This shows that, for AdaGrad under heavy-tailed noise, clipping is a structural stabilizer of the adaptive geometry rather than merely a robustness heuristic.

Machine Learning (cs.LG); Systems and Control (eess.SY)

arXiv:2609.30276 [cs.LG]

(or arXiv:2609.30276v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2609.30276 Focus to learn more arXiv-issued DOI via DataCite

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