Projected stochastic-gradient update

Let $C\subseteq {\mathbb{R}}^d$ be nonempty, closed, and convex, and let $x_0\in C$. The Euclidean projection $P_C(y)$ is the unique nearest point to $y$ in $C$; its existence and equivalence with the proximal map of the indicator of $C$ are supplied by The proximal map of an indicator is Euclidean projection. The indicator is zero on $C$ and $+\mathrm{\infty }$ off $C$, as defined in Extended-real indicator of a set. For the filtration-adapted oracle and deterministic stepsizes ${\alpha }_k>0$ of Filtration-adapted stochastic-gradient oracles and Robbins–Monro updates, a sequence $(x_k{)}_{k\ge 0}$ satisfies the projected stochastic-gradient update when, for every $k\ge 0$, $x_{k+1}=P_C(y_k),y_k=x_k-{\alpha }_k{G}_k\mathrm{.}$

This relation specifies a unique next point $x_{k+1}\in C$ for each $x_k$ and $G_k$, because the projection exists uniquely for every $y_k\in {\mathbb{R}}^d$ under the stated conditions. Its nonexpansiveness is not part of this definition; it is supplied by the proximal-map theorem when used in proofs.