When can Optimally Trained Transformers Achieve Near-Lossless Token Prediction for non-Markovian Data?

Kağan Akman, Jake Feldman Starosta, Naci Saldi, Serdar Yüksel

There exists little mathematical theory explaining why Transformer architectures are empirically successful even for non-Markovian data. To this end, this paper studies next-token prediction via stochastic optimal control theoretically trained Transformers for such sources, where we consider either stationary and ergodic non-Markovian sources or those which are hidden Markov models of some finite order (possibly non-stationary). In either case, we first obtain error bounds via finite models with finite window approximations. For stationary ergodic sources, the resulting error is bounded by an entropy gap; for $N$th order Markovian sources, it decays at an exponential rate $α^{\lfloor M/N\rfloor}$, where $M$ is the window length and $α$ is determined by filter stability (and the Dobrushin coefficients of the transition and measurement kernels). We then construct a Transformer of depth $M+1$, one head per layer, realizing the truncated filter, and we optimally train it with stochastic control developed in earlier work, which rigorously returns a near optimal Transformer. In particular, we show that for a non-Markovian source, either under stationarity and ergodicity for general…