Distributed Optimization via Energy Conservation Laws in Dilated Coordinates

The paper states and analyzes the adaptive symplectic–Euler discretization (Eqs. (15a)–(16c)) and proves an O(1/k^{2−β}) rate under explicit four-case step-size conditions and mild smoothness/convexity assumptions. The proof defines a Lyapunov-like functional Vk, derives a decrement inequality Vk − Vk+1 ≥ 2Ak qk+1 + (sk+1/4)·2θk+1 rk+1, and closes it using that Φk(X) := (2θkh)^{−β}F(X) + 1/2 X^T L̃X is L̃k-smooth, with sk ≤ 2/L̃k ensuring the needed descent; telescoping and treating k=0 via V′0 yield 2Ak(2θkh)^{−β}(F(Xk^+) − F*) ≤ 1/s0||X0 − X*||^2 and thus the advertised bound (25) . By contrast, the model’s alternative Lyapunov E_k replaces the paper’s key distance term ||Z_{k+1}−X*||^2/sk by (h^β/(2sk))||X_k−X*||^2 and asserts a one-step drift inequality in which the crucial rk+1 term is missing the factor sk (paper has θk+1 sk+1/2 · rk+1). This mismatch breaks the negativity argument in the “wk+1 ≤ 0, rk+1 < 0” case unless sk ≥ 2, which is not guaranteed by the step-size rule; hence the claimed monotonicity E_{k+1} ≤ E_k does not follow from the model’s algebra. The paper’s analysis, including the precise L̃-smoothness bound and the use of Vk and V′0, is internally consistent and yields the stated rate, whereas the model’s proof is incorrect on these technical points .