sHAIL-Causal: A Sequential Staircase Procedure for Invariant Causal Predictor Discovery
Abstract
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 arriving sequentially, yielding a confidence guarantee that stays valid at every arrival, which one-shot exhaustive search cannot offer without repeating its full combinatorial search. We close by formalizing the intervention of a wise teacher who lifts a saturated learner off a plateau of boredom.
Related
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BibTeX
@misc{fokoue2026shailcausal,
title = {{sHAIL-Causal: A Sequential Staircase Procedure for Invariant Causal Predictor Discovery}},
author = {Ernest Fokoué},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07057},
eprint = {2610.07057},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.07057},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - sHAIL-Causal: A Sequential Staircase Procedure for Invariant Causal Predictor Discovery
AU - Ernest Fokoué
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.07057
UR - https://arxiv.org/abs/2610.07057
AB - 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 arriving sequentially, yielding a confidence guarantee that stays valid at every arrival, which one-shot exhaustive search cannot offer without repeating its full combinatorial search. We close by formalizing the intervention of a wise teacher who lifts a saturated learner off a plateau of boredom.
ER -