In many first-order optimization methods an oracle is needed whose action enforces the constraint/regularizations. For example, in projected gradient descent, conditional gradient method, and proximal methods, these oracles are the projection oracle, the (constrained) linear optimization oracle, and the prox. operator, respectively.

I could find some papers that analyze the convergence of the first-order methods considering inexact oracles with **additive error**. However, I could not find results on convergence analysis of these methods when we have inexact oracles with **multiplicative error**.

Let's take the projected gradient descent for example. Suppose that we are given an oracle $P_{\mathcal{C},\gamma}(x)$ that returns an approximate projection of any point $x$ onto the compact convex set $\mathcal{C}$ in the sense that

$$ \begin{align*} \|P_{\mathcal{C},\gamma}(x)-x\|&\leq(1+\gamma)\min_{y\in\mathcal{C}}\|y-x\|, \end{align*} $$ for a given $\gamma\geq 0$.

I would appreciate any pointer to references that analyze (approximate) convergence under this kind of oracle model.