Learning in Time-Varying Monotone Network Games with Dynamic Populations

The paper proves almost sure convergence of the randomized, partially-participating projected gradient dynamics to the unique solution of VI(F̃,S) using a projected SA reformulation, a zero-mean bounded-variance noise term, and Robbins–Siegmund’s almost-supermartingale lemma; it then derives almost-sure and expected convergence rates and shows the learned profile is an ε–Nash equilibrium of each realized stage game with high probability. The candidate solution follows the same structure: SA view with diag(P̄) scaling, projection nonexpansiveness, μ-strong monotonicity, an almost-supermartingale one-step inequality, Robbins–Siegmund for a.s. convergence, an induction-based expected-rate under a logistic proxy, and a concentration/Lipschitz argument for ε–Nash. Minor differences (weighted vs Euclidean norm, constants and citations, and a remark about noise variance’s N-scaling) do not affect correctness.