Non-uniform hierarchy theorem for approximating functions
There are some fairly tight results known for tradeoffs between circuit size and theirapproximation, with which one can prove hierarchy theorems for approximating functions. I would start with the following reference:

Alexander E. Andreev, Andrea E. F. Clementi, Jose Rolim. Optimal Bounds for the Approximation of Boolean Functions and Some Applications. ECCC, 1995.