Momentum in large-batch training: Polyak enlarges the critical batch size, Nesterov improves data efficiency

arXiv:2609.02728v1 Announce Type: cross Abstract: We study when and how momentum improves large-batch training in the one-pass regime, using power-law kernel regression as a tractable setting. We first characterize risk stability through the critical learning rate, defined as the largest learning rate for stable training, and obtain $\eta_{\mathrm{SGD}}^{\mathrm{crit}}\eqsim 1$, $\eta_{\mathrm{Polyak}}^{\mathrm{crit}}\eqsim \min\{1,B(1-\rho)\}$, and $\eta_{\mathrm{Nesterov}}^{\mathrm{crit}}\eqsim \min\{1,B^\beta(1-\rho)\}$, where $B$ is the batch size, $\rho$ is the momentum factor, and $\beta>1$ is the capacity exponent. Within this admissible region, we derive scaling laws for the full risk dynamics, capturing the progression from an early transient, through power-law decay, to a noise floor. We then minimize the final-step risk over the admissible learning rates and momentum factors under a fixed data budget, yielding a three-regime batch-size phase diagram that reveals how the role of momentum changes with batch size. Notably, Polyak enlarges the critical batch size, the largest batch size preserving the best small-batch data-scaling exponent, thereby enabling greater parallelism without sacrificing data efficiency. In contrast, Nesterov achieves better data efficiency in the large-batch regime because its look-ahead mechanism suppresses noise accumulation. Numerical experiments validate the predicted stability boundaries, risk dynamics, and batch-size phase diagram.

Abstract:We study when and how momentum improves large-batch training in the one-pass regime, using power-law kernel regression as a tractable setting. We first characterize risk stability through the critical learning rate, defined as the largest learning rate for stable training, and obtain $\eta_{\mathrm{SGD}}^{\mathrm{crit}}\eqsim 1$, $\eta_{\mathrm{Polyak}}^{\mathrm{crit}}\eqsim \min\{1,B(1-\rho)\}$, and $\eta_{\mathrm{Nesterov}}^{\mathrm{crit}}\eqsim \min\{1,B^\beta(1-\rho)\}$, where $B$ is the batch size, $\rho$ is the momentum factor, and $\beta>1$ is the capacity exponent. Within this admissible region, we derive scaling laws for the full risk dynamics, capturing the progression from an early transient, through power-law decay, to a noise floor. We then minimize the final-step risk over the admissible learning rates and momentum factors under a fixed data budget, yielding a three-regime batch-size phase diagram that reveals how the role of momentum changes with batch size. Notably, Polyak enlarges the critical batch size, the largest batch size preserving the best small-batch data-scaling exponent, thereby enabling greater parallelism without sacrificing data efficiency. In contrast, Nesterov achieves better data efficiency in the large-batch regime because its look-ahead mechanism suppresses noise accumulation. Numerical experiments validate the predicted stability boundaries, risk dynamics, and batch-size phase diagram.

Machine Learning (stat.ML); Machine Learning (cs.LG); Optimization and Control (math.OC)

arXiv:2609.02728 [stat.ML]

(or arXiv:2609.02728v1 [stat.ML] for this version)

https://doi.org/10.48550/arXiv.2609.02728 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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Momentum in large-batch training: Polyak enlarges the critical batch size, Nesterov improves data efficiency

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