Scaling Laws for Dynamic Mini-Batch SGD in Sketched Linear Regression

Mini-batching is central to large-scale optimization, yet its role in statistical scaling laws remains limited. We study one-pass and multi-pass batch SGD for sketched linear regression under power-law spectral and source conditions. Our analysis reveals a two-horizon phenomenon induced by warmup--stable--decay schedules: deterministic learning is governed by the full optimization trajectory, while stochastic error retains only a shorter terminal memory. For dynamic batch schedules, the individual batch sizes enter through influence-weighted summaries that measure how strongly each update affects the final risk. Consequently, batching leaves the approximation and optimization-bias laws unchanged at a fixed update horizon, but controls the one-pass variance and the multi-pass fluctuation around full-batch gradient descent. We obtain matching one-pass variance bounds and nearly matching multi-pass fluctuation bounds, recover static-batch and full-batch behavior as special cases, and derive an oracle square-root rule for allocating a fixed iteration budget. These results identify WSD horizon separation and final-risk influence as the mechanisms governing dynamic mini-batch scaling.