Arbitrariness in the definition of characteristic classes

Question

There are many characteristic classes which go by many names but all of them seem to be a function of the curvature two-form $F$ $$f(\mathcal{F})$$ where $f$ has a power-series of the form $$f(x) = 1+a_1 x+a_2 x^2+\cdots$$ Now, it seems to me that these coefficients $a_i$ can be arbitrary. Is there a restriction on $a_i$ for $f$ to be considered a characteristic class? I fail to see how the Chern-Weil theorem may leads to any such restrictions.