Suppose a self-avoiding walk on $\mathbb{Z}^2$, with random steps each one unit long,
backs out of culs-de-sac, but retaining the lattice points on which it stepped
marked as unavailable for future steps.
For example, in (a) below, a cul-de-sac is reached. Unwalking two steps allows
the path to escape and walk in another direction in (b); but note those two
points are marked. Eight steps further another cul-de-sac is reached (c).
And so on.

The walk can continue forever, because once it unwinds to a point
on the bounding box, it is free to step outside that box.
My question is:

What is the growth rate of the distance of the path endpoint from the origin, with respect to the total length of (number of steps in) the path?

Perhaps this model has been studied?
Here's a longer example:

(*Added*.) I can't resist one more, extending the above to 11,371 steps:

(

*Added 29Agu14*.) Permit me to point to the new paper

*j.c.*found and includes in a comment to his (knowledgeable) answer. That paper calls these walks SKSAWs:

*Smart Kinetic Self-Avoiding Walks*.