How Dynamic Batch Schedules Shape Final Prediction Risk

We develop a finite-horizon statistical theory of dynamic mini-batching under warmup–stable–decay learning rates (WSD) for sketched linear regression with power-law spectral and source conditions. WSD induces two effective time scales: deterministic learning is governed by the full horizon $\gamma J$, whereas stochastic noise retains only a shorter terminal memory $\gamma J/\log J$. Although a dynamic batch schedule has one degree of freedom per update, its leading stochastic contribution to final risk collapses to an influence-weighted scalar summary: $B_{\mathrm{eff}}$ for one-pass WSD-SGD and ${\rho }_{\mathrm{eff}}$ under data reuse. This compression also reveals a structural distinction. One-pass batching controls centered SGD variance, while multi-pass batching controls only fluctuation around the full-batch WSD-GD reference; approximation, GD bias, and GD variance remain schedule-independent at a fixed update horizon. We obtain matching one-pass variance bounds and nearly matching multi-pass fluctuation bounds. As a consequence, minimizing the influence-weighted stochastic terms yields the oracle allocation $B_t\propto \sqrt{{\kappa }_t}$.