Convergence and Stability Analysis of a Generalized Proximal Point Algorithm and Its Inexact Version

What the math gives to ML

The paper replaces ordinary monotonicity of an operator F with monotonicity of a pair (F,v), where v can be selected to make a difficult nonmonotone problem tractable and may have an easier inverse. This suggests a preconditioned implicit optimizer: solve an approximate generalized proximal step in a learned or diagonal metric v rather than taking an explicit gradient step. The transferable asset is the pair inequality \(\langle F(x)-F(y),v(x)-v(y)\rangle\geq\alpha\|x-y\|^2\), which can serve both as a stability criterion and as a way to tune the preconditioner. The main caveat is that each step requires an inner solve, so the method should first be tested on small models where stability or larger outer-step sizes can compensate for the inner-loop cost.