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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 duelingfeedback returns a (noisy) single-bit binary comparison of the function valuesof the two queried points. The translation of the function values to the singlecomparison bit is through a \emph{transfer function}. This problem has beenaddressed previously for some restricted classes of transfer functions, buthere we consider a very general transfer function class which includes allfunctions 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 theobjective is smooth and strongly convex.

Dueling Convex Optimization with General Preferences