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The proximal gradient method minimizes $g+h$, where $g$ has an $L$-Lipschitz gradient and $h$ is convex with a tractable proximal operator, by $x_{r+1}={\mathrm{prox}}_{\alpha h}(x_r-\alpha \nabla g(x_r))$. A standard choice $0<\alpha \le 1/L$ gives objective error $O(1/r)$ in the general convex case.

This basic forward-backward update is one of the proximal gradient methods for learning.