sHAIL-Causal: A Sequential Staircase Procedure for Invariant Causal Predictor Discovery

Ernest Fokoué

We introduce sHAIL-Causal, the causal specialization of the Saturated Hierarchical Atomic Incremental Learning (sHAIL) paradigm: a sequential staircase procedure that ascends a nested hierarchy of hypothesis classes H_0 < H_1 < ... < H_K once a saturation signal indicates that mastery of the current stage has plateaued. Where general sHAIL leaves the saturation criterion open, sHAIL-Causal instantiates it with a joint criterion of goodness-of-fit saturation and cross-environment invariance, replacing the complexity control of Structural Risk Minimization. We show, theoretically and by simulation, that complexity-only staircases are seduced by confounded predictors that lower empirical risk without reflecting stable causal structure, whereas an invariance-gated staircase provably halts at the true causal predictor set under a per-variable Richness condition. We show that naive greedy search fails to recover the causal set even under Richness, trace the failure to non-monotonicity of the invariance statistic along single-variable paths, and validate a fix combining bounded-exhaustive block-seeding with a calibrated acceptance threshold. We then extend the guarantee to environments…