The Impossible Trinity of Time-Series Validation: A Conservation Law among Training Sufficiency, Test Coverage, and Temporal Causality

arXiv:2609.29530v1 Announce Type: new Abstract: Validating a model on a time series asks for three things at once: each training run should use most of the sample (sufficiency), the test sets should together cover most of the sample (coverage), and training data should come before test data (causality). We prove that the three cannot be had together and price each one. Let alpha be the smallest training fraction over folds, eta the fraction of the sample covered by tests, Lambda the fraction of the sample used as training data from the future of a test point, and elta the distance from a test point to the nearest training point in its future. Every scheme on a sample of length T satisfies alpha+eta le 1+Lambda and alpha+min{eta,elta/T} le 1, and under eta-mixing the leakage bias at a test point is at most 2Meta_{mix}(elta). In words: going beyond the causal frontier alpha+eta=1 requires training on the future; that future data must sit within (1-alpha)T of a test point; and its harm depends on its distance, not its amount. Hence expanding walk-forward is exactly the Pareto frontier of causal validation, k-fold cross-validation buys the most future data, and purged k-fold with an embargo pays in distance instead, which is cheap when the process forgets quickly but cannot repair the part of causality demanded by non-stationarity. On pure noise, shuffled 5-fold reports an information coefficient of +0.32, while contiguous 5-fold, using the same amount of future data, reports +0.004.

Source: arXiv cs.LG | 2026-09-25