Adaptive Batch Sizes Using Non-Euclidean Gradient Noise Scales for Stochastic Sign and Spectral Descent

TL;DR

This paper introduces a non-Euclidean gradient noise scale for adaptive batch sizing tailored to sign-based and spectral descent optimizers, improving training efficiency and reducing steps needed.

Contribution

It derives gradient noise scales suited for non-Euclidean geometries and proposes an efficient estimation method for practical adaptive batch size adjustment.

Findings

Achieves up to 66% reduction in training steps.

Matches validation loss of constant-batch baselines.

Effective on large-scale language models.

Abstract

To maximize hardware utilization, modern machine learning systems typically employ large constant or manually tuned batch size schedules, relying on heuristics that are brittle and costly to tune. Existing adaptive strategies based on gradient noise scale (GNS) offer a principled alternative. However, their assumption of SGD's Euclidean geometry creates a fundamental mismatch with popular optimizers based on generalized norms, such as signSGD / Signum ($\ell_\infty$) and stochastic spectral descent (specSGD) / Muon ($\mathcal{S}_\infty$). In this work, we derive gradient noise scales for signSGD and specSGD that naturally emerge from the geometry of their respective dual norms. To practically estimate these non-Euclidean metrics, we propose an efficient variance estimation procedure that leverages the local mini-batch gradients on different ranks in distributed data-parallel systems. Our experiments demonstrate that adaptive batch size strategies using non-Euclidean GNS enable us to match the validation loss of constant-batch baselines while reducing training steps by up to 66% for Signum and Muon on a 160 million parameter Llama model.