From Adam to Adam-Like Lagrangians: Second-Order Nonlocal Dynamics
This work provides a theoretical framework for understanding Adam's dynamics, which is incremental for optimization researchers.
The paper tackled the problem of modeling the Adam optimizer as a continuous-time dynamical system by deriving an accelerated second-order integro-differential formulation, with numerical simulations on Rosenbrock-type examples showing agreement between the proposed dynamics and discrete Adam.
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.