Transformer-based language models exhibit an astonishing capacity to generalize to longer and more difficult problem instances than those seen in training. However, theoretical understanding of this phenomenon still lags behind. Unlike most machine learning models, the generalisation abilities of transformers inherently require considering two axes. On one hand, one must examine in-distribution statistical generalisation, which concerns the ability of a model to broaden a function from training samples to novel samples of the same distribution. On the other hand, given the sequential nature of language data, one must also consider length generalisation, which reflects whether a model trained on inputs of a given length can extrapolate to longer ones. These two notions have mostly been studied separately, leaving open how they interact. In this work, we give a joint treatment of both axes for fixed-precision transformers. We give generalisation guarantees in terms of the parameter sizes of the involved models, as well as the quantization space and the alphabet size that supply the sequences. Finally, we illustrate how the requirements in the sample count and training length trade off. In particular, we find that increasing training length can be exponentially more efficient than increasing the sample size. Experiments support predictions of the theory.

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