How do mirror prox and accelerated methods differ conceptually? In the very basic case of nice function on Hilbert space, dual space is same as primal, it seems to me that they to almost same thing: at point $x_t$ they apply a gradient obtained by looking "one step ahead" $x_{t+1} = x_t + \eta \nabla f(x_t + \eta \nabla f(x_t))$. Mirror prox adds an additional restriction on closeness of both "look ahead" and "gradient step" points in terms of some divergence:
$$ \hat z_t = \arg\min_z (D(z, z_t) + \langle \eta \nabla f(z_t), z \rangle)$$
$$ z_{t+1} = \arg\min_z (D(z, z_t) + \langle \eta \nabla f(\hat z_t), z \rangle)$$
Does the divergence-closeness part is the only difference? Should they perform similarly in saddle point problems without constraints?
Thank you.

I've never seen it referred to as a "mirror prox", however I know that D-resolvents can be viewed as a generalization of a proximal operator for when you re-define your distance in a Hilbert space. These notions of distance are also known as "Bregman divergences," and capture the KL-divergence as a special case. When you use the distance $\|z-z_t\|^2/2$, the D-resolvent reduces to the "standard" proximal operator. All of the basic properties for proximal operators have analogies for D-resolvents, and THE paper on this subject is by Bauschke, Borwein and Combettes. Here is a free version.
Also, examples of proxes for when you redefine your distance can be found, e.g., here.
