Deterministic Adam-Inspired Methods with Accelerated Convergence Rate

Adam is widely used, but its convergence theory remains incomplete even in the deterministic full-batch setting because momentum and adaptive preconditioning are tightly coupled. For smooth convex objectives, we split the momentum variable through variable-and-operator splitting, which reveals the acceleration mechanism. We then combine a Hessian-driven correction with Adam-style feedback based on the gradient magnitude. The resulting Adam-HNAG (Hessian-driven Nesterov accelerated gradient with Adam-style adaptive preconditioning) flow admits a nonnegative energy that decays exponentially. Its discretization yields two methods, Adam-HNAG and the synchronous variant Adam-HNAG-s. Under the stated trajectory-bound and consistency conditions, both methods satisfy a discrete Lyapunov contraction. If the exact adaptive steps are accepted, this contraction gives an $O(k^{-2})$ objective-value bound. Numerical experiments illustrate their behavior. These results apply to the proposed methods, not to the original Adam recursion.