Training Generative Adversarial Networks with Adaptive Composite Gradient

Proposition 5.5 asserts linear convergence under the parameter conditions 0 < α+β1 ≤ 1/√λmax(ATA) and |α+β1| + |2β1(1+γ)| ≤ (α+β1)²√λmin(ATA)/10 for nonsingular A (equation (18)). Setting x := α+β1 > 0 and σmin := √λmin(ATA), σmax := √λmax(ATA), the second inequality implies x ≤ x²σmin/10, hence x ≥ 10/σmin, which contradicts x ≤ 1/σmax unless σmax ≤ σmin/10. Since σmax ≥ σmin > 0 for nonsingular A, the feasible set is empty. Thus the stated linear convergence is (at best) vacuous and the theorem, as written, provides no nonempty parameter regime. This directly matches the model’s argument and contradicts the intended content of Proposition 5.5. See the paper’s update rule (15) and iterative matrix (16), the spectral factorization (Proposition 5.4), and the stated conditions (18) for confirmation . The appendix’s subsequent spectral-radius manipulations do not repair the impossibility and even include an imprecise remark that τ is “almost small,” which is not a substitute for a valid parameter regime . The paper’s conclusion also overstates by claiming a linear convergence rate in general, which is unsupported under the stated (infeasible) conditions .