What One Cannot, Two Can: Two-Layer Transformers Provably Represent Induction Heads on Any-Order Markov Chains
This addresses a fundamental theoretical gap in understanding transformer capabilities for structured sequence modeling, though it is incremental as it builds on prior work on induction heads and Markov processes.
The paper tackles the problem of whether a two-layer transformer with one head per layer can represent any kth-order Markov process for in-context learning, and it proves theoretically that such a construction is possible, providing the tightest known characterization of transformer depth and Markov order.
In-context learning (ICL) is a hallmark capability of transformers, through which trained models learn to adapt to new tasks by leveraging information from the input context. Prior work has shown that ICL emerges in transformers due to the presence of special circuits called induction heads. Given the equivalence between induction heads and conditional k-grams, a recent line of work modeling sequential inputs as Markov processes has revealed the fundamental impact of model depth on its ICL capabilities: while a two-layer transformer can efficiently represent a conditional 1-gram model, its single-layer counterpart cannot solve the task unless it is exponentially large. However, for higher order Markov sources, the best known constructions require at least three layers (each with a single attention head) - leaving open the question: can a two-layer single-head transformer represent any kth-order Markov process? In this paper, we precisely address this and theoretically show that a two-layer transformer with one head per layer can indeed represent any conditional k-gram. Thus, our result provides the tightest known characterization of the interplay between transformer depth and Markov order for ICL. Building on this, we further analyze the learning dynamics of our two-layer construction, focusing on a simplified variant for first-order Markov chains, illustrating how effective in-context representations emerge during training. Together, these results deepen our current understanding of transformer-based ICL and illustrate how even shallow architectures can surprisingly exhibit strong ICL capabilities on structured sequence modeling tasks.