Semi-Gradient SARSA Routing with Theoretical Guarantee on Traffic Stability and Weight Convergence

The paper’s formal theorem establishes sufficiency (convergence when λ < Σ μn) but does not actually prove the necessity (“only if”) direction it repeatedly claims; moreover, key steps in the proof rely on extra conditions (e.g., ‘sufficiently small’ softmax temperature and a small Lipschitz constant for the policy) that are not stated in Theorem 1 and appear only inside lemmas or proofs. See the theorem statement and surrounding exposition, which assume λ < Σ μn and then prove convergence, while the text elsewhere asserts an iff claim without a corresponding converse proof; and note the additional small-ι and small-Lπ requirements used in Lemma 1 and the ODE argument, respectively . The candidate model gives both directions, but its sufficiency proof depends on unsubstantiated steps (e.g., a uniform Foster–Lyapunov drift for all frozen policies, and identifying the SA mean field with a gradient under the optimal stationary distribution), so it is also incomplete.