Dry-Friction Inertial Dynamics with Implicit Hessian-Driven Damping: Finite-Time Stabilization, Shadowing, and Proximal Discretization

What the math gives to ML

The paper combines viscous damping, nonsmooth dry friction, and a look-ahead gradient in an inertial optimization inclusion. Its most transferable construction is an optimizer whose velocity is updated through an exact proximal operator for a dry-friction penalty, while the gradient is evaluated at a shifted parameter point that approximates Hessian damping without explicitly forming a Hessian. For Euclidean friction, the proximal step is vector soft-thresholding and can set velocity exactly to zero when the force is sufficiently small, potentially reducing late-training oscillations. The implementation should be tested as a momentum optimizer against SGD with momentum, Nesterov momentum, and AdamW, measuring both convergence and the frequency of exactly stationary updates.