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

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 $η_{\mathrm{SGD}}^{\mathrm{crit}}\eqsim 1$, $η_{\mathrm{Polyak}}^{\mathrm{crit}}\eqsim \min\{1,B(1-ρ)\}$, and $η_{\mathrm{Nesterov}}^{\mathrm{crit}}\eqsim \min\{1,B^β(1-ρ)\}$, where $B$ is the batch size, $ρ$ is the momentum factor, and $β>1$ 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.

Spectral Gradient Orthogonalization Improves Differentially Private Training at Scale

Differentially private training adds isotropic Gaussian noise to clipped gradients, corrupting every singular direction equally. In vision models, where spatial correlation concentrates gradient energy into a low-rank subspace, most of this noise falls in directions that carry little signal. Spectral gradient orthogonalization via polar decomposition is introduced as a post-processing step that recovers directional signal from the noisy gradient's low-rank structure at zero additional privacy cost. A phase transition governs the utility of this approach: orthogonalization improves accuracy only when the per-direction spectral signal-to-noise ratio (SNR) suffices for singular vector recovery; in low-SNR regimes, the directional bias of the gradient is replaced by a nearly random orthogonal update, and the transformation is harmful. The recovery threshold is determined by the spectral gap of the gradient and is surpassed at large batch sizes. Empirically, the benefit scales with model capacity: spectral orthogonalization achieves a +20.9% improvement over DP-SGD on WRN-28-10 (B = 4096) and +14.9% on ResNet-18, while reducing inter-run variance by a factor of two to three. In the fine-tuning regime, spectral orthogonalization matches the stability of DP-Adam while maintaining a first-order memory footprint. Combining spectral with temporal denoising yields 50.3% on CIFAR-10 (epsilon = 4), the highest accuracy in any tested configuration. These gains are specific to moderate-to-high-SNR regimes such as large-batch training of higher-capacity models. Small-batch or low-SNR settings are better served by DP-SGD or temporal denoising.