q0: Primitives for Hyper-Epoch Pretraining

Multi-epoch training is becoming the standard now that compute is growing faster than the supply of high-quality text. But pretraining a single model saturates within a few passes, long before the compute budget is exhausted. We argue this calls for a conceptual shift from training a single model toward exploring a population of models and aggregating their predictions. We introduce hyper-epoch pretraining (q0), which turns a multi-epoch budget into a population of diverse models whose combined predictions reach a lower validation loss than a single refined model. q0 reduces to three core primitives. A cyclic schedule with anti-correlated learning rate and weight decay collects diverse models from a few parallel trajectories. Chain distillation trains each model against its predecessor so that model quality compounds across the population. A learned prior, fit on a held out set, selects and weights members for any inference budget. On a 1.8B-parameter model trained on 100M FineWeb tokens, q0 matches a strong 256-epoch ensemble baseline using only ~56 epochs (~4.6x fewer), or ~67 epochs (~3.8x fewer) when matched to the baseline's ensemble size, and continues to improve beyond it. These gains reach cumulative ~12.9x data efficiency under the Slowrun setting and transfer to downstream benchmarks. Crucially, the optimal allocation shifts with the budget, so we give prescriptive recipes for how to spend a given epoch budget to maximize generalization, from a single epoch up to the largest budgets.