One-Layer Transformers are Provably Optimal for In-context Reasoning and Distributional Association Learning in Next-Token Prediction Tasks

We study the approximation capabilities and on-convergence behaviors of one-layer transformers on the noiseless and noisy in-context reasoning of next-token prediction. Existing theoretical results focus on understanding the in-context reasoning behaviors for either the first gradient step or when the number of samples is infinite. Furthermore, no convergence rates nor generalization abilities were known. Our work addresses these gaps by showing that there exists a class of one-layer transformers that are provably Bayes-optimal with both linear and ReLU attention. When being trained with gradient descent, we show via a finite-sample analysis that the expected loss of these transformers converges at linear rate to the Bayes risk. Moreover, we prove that the trained models generalize to unseen samples as well as exhibit learning behaviors that were empirically observed in previous works. Our theoretical findings are further supported by extensive empirical validations.