# Optimization of strongly convex functions with approximate evaluations of gradient and Hessian

Suppose I want to find the minimum of a differentiable, strongly convex function $$f:\mathbb{R}^n\to\mathbb{R}$$ with constant $$\mu>0$$. That is, for all $$x,y\in\mathbb{R}^n$$, I have that: $$f(y) \geq f(x)+\nabla f(x)^{T}(y-x)+\frac{\mu}{2}\|y-x\|^{2}.$$ If I can efficiently evaluate $$f$$ and its gradient, then it is clear how to find the minimum of $$f$$ with first-order methods. But now assume that I only have access to approximate entries of $$\nabla f(x)$$, say up to an additive error $$\epsilon$$. What is the state of the art result on how small $$\epsilon$$ has to be compared to $$\mu$$ so that I can still ensure convergence of some optimization method to an approximate minimum?

For the problem you describe, since the function is strongly convex and $$x,y \in \mathbb{R}$$ (so you don't have bound constraints), your solution is obtained by solving the system $$\nabla f = 0$$.
Now, if you only have approximations to the gradient, the resulting error largely depends on the algorithm you chose to use to solve this. For instance, since your function remains strongly convex, $$\mu$$ and $$\epsilon$$ would not affect each other at all if you simply solve the nonlinear system $$\nabla f = 0$$.