Sharp bounds for stochastic proximal and projection estimators via radial dominance

What the math gives to ML

The paper provides a constructive zeroth-order approximation to proximal maps and metric projections: perturb an input with Gaussian noise, exponentially reweight samples by an energy, and return their barycenter. Its transferable asset is the covariance formula for the Jacobian and the resulting cocoercivity guarantee for weakly convex energies, which give a principled stability bound for a stochastic neural layer. This can replace an expensive implicit proximal solve with a small Monte Carlo module using only function evaluations. The strongest applications are energy-based activations and differentiable constraint layers where exact proximal computation is unavailable or costly.