  We address the problem of \emph{convex optimization with dueling feedback},
where the goal is to minimize a convex function given a weaker form of
\emph{dueling} feedback. Each query consists of two points and the dueling
feedback returns a (noisy) single-bit binary comparison of the function values
of the two queried points. The translation of the function values to the single
comparison bit is through a \emph{transfer function}. This problem has been
addressed previously for some restricted classes of transfer functions, but
here we consider a very general transfer function class which includes all
functions that can be approximated by a finite polynomial with a minimal degree
$p$. Our main contribution is an efficient algorithm with convergence rate of
$\smash{\widetilde O}(\epsilon^{-4p})$ for a smooth convex objective function,
and an optimal rate of $\smash{\widetilde O}(\epsilon^{-2p})$ when the
objective is smooth and strongly convex.
