It is shown that even a three-token, single-layer attention-only transformer can fail to converge, and that, while each transformer has a temperature threshold, the supremum of this set is infinite, so there is no global minimal temperature that forces all transformers to converge.
Abstract
In this paper we study the behavior of transformer networks generating text over long periods of time. We show that even a three-token, single-layer attention-only transformer can fail to converge. We then study a rich class of transformers with arbitrary finite depth, layer normalization, MLPs, RoPE encodings, and an arbitrary mix of local and global layers. Every transformer in this class has a softmax temperature threshold above which its empirical cache distribution converges to a unique limit, independent of initial conditions. We show further that, while each transformer has a temperature threshold, the supremum of this set is infinite, so there is no global minimal temperature that forces all transformers to converge.
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