The Geometry of Algorithmic Stability: A Hodge Theoretic View on Structural vs. Statistical Instability

Abstract

Algorithmic stability—the robustness of predictions to training data perturbations—is fundamental to reliable machine learning. We propose a unified mathematical framework that rigorously distinguishes between two sources of instability: structural inconsistency and statistical variance. We formalize structural inconsistency using Combinatorial Hodge Theory, characterizing it as cyclical flows (Condorcet cycles) on a graph of hypotheses. This framework reveals that methods like inflated operators and regularization specifically target these structural obstructions, while methods like bagging primarily address statistical variance. We provide direct empirical validation through three key experiments. First, in a controlled setting with engineered Condorcet cycles (pure structural instability), inflated operators achieve perfect stability while bagging fails, confirming the core distinction. Second, we validate on a standard digit classification task that structural obstructions are negligible ($||C_{cycle}|| \approx 2.3 \times 10^{-16}$, machine precision), explaining the empirical dominance of variance-reduction methods. Third, we demonstrate that significant structural obstructions naturally emerge in fairness-constrained model selection on real-world data ($||C_{cycle}|| = 0.857$, approximately $10^{15}$ times larger), providing a topological characterization of the instability arising from incompatible objectives.