Generalized Lyapunov exponents and aspects of the theory of deep learning

The paper states the noncommutative ergodic theorem for nonexpansive cocycles on weak metric spaces (Theorem 3) and sketches the proof by (i) verifying subadditivity of a(n,ω)=d(x0,u(n,ω)x0) and invoking Kingman to obtain the drift ℓ, and (ii) following the horofunction/metric-functional argument of Gouëzel–Karlsson (2020) to build h_ω so that −h_ω(u(n,ω)x)/n converges to the same ℓ, with a key “good times” refinement of subadditivity used for the lower bound. This is explicitly described around the subadditivity and Kingman limit discussion and in the statement of Theorem 3, together with the note that the proof follows [GK20, §3] in the weak metric setting . The candidate solution reproduces exactly this proof strategy: Kingman for the rate of escape, independence of x, compactness of the metric compactification, a “good times” subsequence, and the Lipschitz upper bound for all metric functionals, yielding equality of limits for a suitably chosen h_ω. The technical steps align with the paper’s sketch and the GK20 method (also visible in the linear-operator application where the same good-times inequality is used) . Minor differences are purely expository (e.g., the model attributes the good-times lemma to KL06; the paper emphasizes the stronger GK20 refinement).