Computational Inertia as a Conserved Quantity in Frictionless and Damped Learning Dynamics
This work provides a theoretical tool for analyzing convergence, stability, and training geometry in machine learning, but it is incremental as it builds on existing optimization dynamics concepts.
The paper identified a conserved quantity called computational inertia in continuous-time optimization dynamics, defined as the sum of kinetic and potential energy, which remains invariant under frictionless training and decays under damping and stochastic perturbations, as demonstrated in a synthetic system.
We identify a conserved quantity in continuous-time optimization dynamics, termed computational inertia. Defined as the sum of kinetic energy (parameter velocity) and potential energy (loss), this scalar remains invariant under idealized, frictionless training. We formalize this conservation law, derive its analytic decay under damping and stochastic perturbations, and demonstrate its behavior in a synthetic system. The invariant offers a compact lens for interpreting learning trajectories, and may inform theoretical tools for analyzing convergence, stability, and training geometry.