Stabilization Limits of Payoff-Based Higher-Order Replicator Dynamics

What the math gives to ML

The paper provides a nontrivial passivity-based stability mechanism for higher-order replicator learning: a softmax state is driven by an auxiliary LTI payoff filter, and strict passivity of that filter is sufficient for convergence in contractive games. Its converse is especially transferable: any nonpassive auxiliary filter can destabilize a suitable strictly contractive game, so adding momentum or higher-order state is not harmless and has a checkable frequency-domain boundary. A neural-network analogue is a simplex-constrained optimizer or attention-routing module whose logits receive gradients through a designed passive LTI filter, with the filter certified by a KYP inequality and monitored during training.