Why Does Adaptive Batching Help LLM Pretraining? A Perspective from Unbounded Variance

Arda Fazla, Antesh Upadhyay, Ege C. Kaya, M. Berk Sahin, Abolfazl Hashemi

Increasing the batch size during training is a common practice in large language model (LLM) pretraining, yet the theoretical justification behind its success is not well understood. Analyses of stochastic optimization often assume uniformly bounded stochastic gradient variance, yet recent evidence suggests that this assumption fails in many practical nonconvex problems. The Blum--Gladyshev (BG-$0$) noise model relaxes this assumption by allowing the variance to grow quadratically with the distance from initialization, suggesting that batch size schedulers can help by controlling the variance growth during training. However, this growth can be overly conservative in practice. We empirically investigate variance growth in LLM pretraining and observe that a generalized BG model with a tunable growth exponent provides a tighter description of practical noise behavior. Motivated by this observation, we introduce the generalized BG-$a$ noise model, which interpolates between bounded variance ($a=0$) and BG-$0$ noise ($a=2$). Under $L$-smoothness, we derive an information-theoretic lower bound with growth-dependent oracle complexity $Ω(ε^{-(4+a)})$ and establish a matching upper bound…