Learning Nonlinear Couplings in Network of Agents from a Single Sample Trajectory

The candidate solution proves exactly the high‑probability oracle inequality stated as Theorem 3 in the paper, including the error decomposition E(ψ) = (1/Ne)(Eπ||Fψ−φ||^2 + σ^2) (equation (26)), the coercivity-based strong convexity (Assumption 5 and inequalities (36)–(38)), uniform Lipschitz/envelope control for LT(ψ) (equation (32)), and the geometric-ergodicity concentration culminating in the same sample-size condition and bound as (39)–(40) (via Lemma 3/Proposition 2). The candidate’s route uses a classical uniform deviation + ERM argument, while the paper’s proof uses a ratio-of-excess-risk device, but the constants, dependence on cH, K, R, S, and the covering number N(H,·) all match the paper’s statement. Minor differences (e.g., constants in Lipschitz bounds, a brief extraneous formula for R, and K vs K^2 in an intermediate upper bound) do not affect the final claim, which coincides with (39)–(40). Overall, the paper’s argument and the candidate’s solution are consistent and correct, with substantially similar ingredients and conclusions. See (26) for the risk identity, (36)–(38) for coercivity, (32) for uniform continuity/envelope, and Theorem 3 for the oracle inequality and sample-size condition .