From Adam to Adam-Like Lagrangians: Second-Order Nonlocal Dynamics

In this paper, we derive an accelerated continuous-time formulation of Adam by modeling it as a second-order integro-differential dynamical system. We relate this inertial nonlocal model to an existing first-order nonlocal Adam flow through an $α$-refinement limit, and we provide Lyapunov-based stability and convergence analyses. We also introduce an Adam-inspired nonlocal Lagrangian formulation, offering a variational viewpoint. Numerical simulations on Rosenbrock-type examples show agreement between the proposed dynamics and discrete Adam.

Modeling AdaGrad, RMSProp, and Adam with Integro-Differential Equations

In this paper, we propose a continuous-time formulation for the AdaGrad, RMSProp, and Adam optimization algorithms by modeling them as first-order integro-differential equations. We perform numerical simulations of these equations to demonstrate their validity as accurate approximations of the original algorithms. Our results indicate a strong agreement between the behavior of the continuous-time models and the discrete implementations, thus providing a new perspective on the theoretical understanding of adaptive optimization methods.