Approximating convex functions via non-convex oracles under the relative noise model

Abstract

We study succinct representations of a convex univariate function φ over a finite domain. We show how to construct a succinct representation, namely a piecewise-linear function φ̄ approximating φ when given a black box access to an L-approximation oracle φ̃ of φ (the oracle value is always within a multiplicative factor L from the true value). The piecewise linear function φ̄ has few breakpoints (poly-logarithmic in the size of the domain and the function values) and approximates the true function φ up to a (1+ϵ)L2 multiplicative factor point-wise, for any ϵ>0. This function φ̄ is also convex so it can be used as a replacement for the original function and be plugged in algorithms in a black box fashion. Finally, we give positive and negative results for multivariate convex functions.