optimal solution for expected absolute loss function [on hold]

optimal solution for expected absolute loss function [on hold]

guys,
I need to calculate the optimal solution f(x) for the expected absolute
loss function, anyone know how to solve it? thank you so much!
let's consider a similar problem first: for squared loss function
$$loss(f(x), y) = (f(x) - y)^2$$ the expected loss function would be
$$E[loss(f(x),y)] = \int\int (f(x)-y)^2 p(x,y)dxdy$$
To minimize the expected loss function, we can do as the following way:
$$\int\int (f(x)-y)^2 p(x,y)dxdy \\= \int\int\big \{(f(x) -
E[y|x])^2+(y-E[y|x])^2 +2(f(x)-E[y|x])(E[y|x]-y)\big\}p(x,y)dxdy $$ Since
$$\int\int 2(f(x)-E[y|x])~(E[y|x]-y)~p(x,y)dxdy = 0$$ it could be proved
that when $$f(x)=E[y|x]$$ we will minimize $$E[loss(f(x),y)]$$
now, what if we define? $$loss(f(x),y) = \big |~f(x)-y~ \big |$$
what would be the optimal $f(x)$?