Online Optimization in Games via Control Theory: Connecting Regret, Passivity and Poincaré Recurrence

The paper’s Theorem 18 establishes Poincaré recurrence for dynamical game systems where each agent uses a convex combination of FTRL and the underlying game is a graphical constant-sum game with a fully mixed Nash equilibrium. The proof proceeds by (i) showing the score-space ODE is divergence-free (volume preserving) and (ii) using a constant-of-motion whose level sets are bounded after the q → q′ reduction; this yields recurrence in R^{n−1}, and continuity of the conversion map transfers recurrence to Δ . By contrast, the model’s solution relies on two incorrect or unproven steps: (a) it asserts that a convex combination of FTRL maps reduces to a single FTRL map via an aggregated regularizer with a globally C^1-diffeomorphic mirror map; this conflicts with the paper’s own remark that such combinations need not be FTRL (they could not find a regularizer for ½·RD+½·OGD) and, in particular, fails for non-barrier regularizers like L2 where the mirror map need not be a diffeomorphism onto int Δ ; (b) it concludes Lebesgue-a.e. recurrence in Δ by claiming the push-forward invariant measure is equivalent to Lebesgue via the Jacobian of the mirror map—this equivalence need not hold when the conversion map is not a diffeomorphism (e.g., OGD has boundary contacts). The core volume-preservation and bounded-level-set arguments align with the paper, but the model’s added diffeomorphism/absolute continuity claims are unjustified.