arXiv Machine Learning By Kimia Kazemian, Menghan Xu, John Thickstun, Sarah Dean

Spatial Induction Heads: In-Context Learning of Multidimensional Cellular Automata

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The paper introduces spatial induction heads, a two‑layer gather‑and‑match mechanism that enables transformers to perform in‑context learning on multidimensional cellular automata. By reconstructing local spatial neighborhoods and matching them against prior occurrences, the method achieves Bayesian counting and can approximate the Bayesian posterior for stochastic rules or predict deterministically for deterministic rules. Experiments show that trained transformers generalize to unseen rules in one and two dimensions, achieving near‑perfect rollouts and KL divergences below 0.005 nats from the Bayes‑optimal predictor, with attention patterns supporting the proposed computation.

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