Adapting, Fast and Slow: On Few-Shot Transportability of Compositions

Generalization across domains requires stable structure that links the source and target distributions. Building on causal transportability theory, we study a sequential prediction setting in which the target predictor can be represented as a circuit composed of causal mechanisms that are learnable from source data. We introduce two classes of transportability. Module transportability captures the atomic case, where the target predictor is given by a mechanism learnable from a single source domain. Circuit transportability generalizes this idea to target predictors obtained by composing several modules learned from source data, enabling zero-shot prediction even when no source mechanism directly predicts the target label. We study these classes of circuits under increasingly relaxed assumptions. First, we provide conditions under which the relevant circuits can be learned from source data alone, given causal knowledge about the source and target domains. We then relax these structural assumptions by allowing limited data from the target domain. In particular, we develop a supervised domain adaptation scheme that learns circuits without requiring explicit causal structure. The resulting few-shot guarantees tie the achievable error to the size of the smallest target circuit composable from modules learned from source data. Finally, we propose a gradient-based relaxation of the symbolic circuit search and evaluate it empirically, showing that it qualitatively tracks the predicted regimes of fast adaptation -- with and without process supervision over intermediate positions -- and slow adaptation when no source mechanism matches.