Natural Invariant Measures for Chaotic Game Dynamics: Finding Order in Chaos

What the math gives to ML

The paper’s transferable mechanism is that a chaotic discrete learning map can remain statistically predictable through its natural invariant measure even when individual trajectories do not converge. The measure is estimated from long-run empirical occupations of the orbit, and its integrals provide asymptotic averages of arbitrary observables such as payoff, cost, or regret. A neural-network analogue is to treat optimizer states or recurrent hidden states as an iterated dynamical system, estimate their occupation measure online, and monitor statistical stationarity rather than demanding pointwise convergence. The most practical first application is a chaos-aware training monitor or learning-rate controller that distinguishes stable, periodic, and chaotic regimes using convergence of empirical measures.