A Refined Parameter Condition in the Lyapunov Analysis of IGAHD

What the math gives to ML

The paper gives a sharper Lyapunov-stability region for inertial gradient descent with Hessian-driven damping by retaining exact coefficients in a positive-definiteness argument. The transferable asset is a computable coupling between step size, smoothness estimate, and gradient-difference damping strength, which permits damping settings excluded by the older condition and remains valid for sufficiently small steps. A practical adaptation is an IGAHD optimizer with online smoothness estimation and a parameter controller that enforces the refined inequality while maximizing the admissible step size. The quadratic spectral analysis also supplies a useful diagnostic: on nearly quadratic problems, stable training should exhibit geometric decay rather than merely nonincreasing loss.