Causal Optimizer Interaction Calculus: Hidden Geometric Relaxation and Identifiable Interventions

What the math gives to ML

The paper supplies a causal calculus for separating genuine interactions between optimizer mechanisms from artifacts of parameterization, stochasticity, or incomplete intervention designs. Its most transferable asset is the Möbius decomposition of responses over a lower-finite intervention poset, together with the incidence-operator characterization of which interaction effects are actually identifiable. In neural-network research, this can become an optimizer and training-dynamics audit: run controlled factorial interventions on momentum, adaptivity, clipping, weight decay, schedules, or data ordering, then recover pure higher-order effects and predict untested configurations. The result is a principled alternative to one-factor-at-a-time ablations and can guide removal or combination of optimizer components.