TITLE: Optimal step size in gradient descent
QUESTION [18 upvotes]: Suppose a differentiable, convex function $F(x)$ exists. Then $b = a - \gamma\nabla F(a)$ implies that $F(b) \leq F(a)$ given $\gamma$ is chosen properly. The goal is to find the optimal $\gamma$ at each step. In my book, in order to do this, one should minimize  $G(\gamma)=$ $F(x-\gamma\nabla F(x))$ for $\gamma$. It also says that it should be minimized via a line search. Why can't this function be minimized by simple calculus?
I am unsure what it means to perform a line search on this function. I am also unsure why we want to minimize this function to find the optimal step size.

REPLY [2 votes]: Not your question,
but an adaptive step size can beat a constant $\gamma$, 
and a $\gamma_i$ per component can beat a single $\gamma$ for all components.
See RmsProp ,
rmsprop.py
and Zeiler, ADADELTA: An adaptive learning rate method, 2012, 6p.
But be careful with recursive / IIR filters !