Approximating Convex Functions By Non-Convex Oracles Under The Relative Noise Model

We study succinct approximation of functions that have noisy oracle access. Namely, construction of a succinct representation of a function, given oracle access to an L-approximation of the function, rather than to the function itself. Specifically, we consider the question of the succinct representation of an approximation of a convex function φ that cannot be accessed directly, but only via oracle calls to a general (i.e., not necessarily convex) L-approximation φ̃ of φ. We efficiently construct such a succinct (1+ ε)L-approximation for a univariate convex φ, for any ε > 0. The algorithms designed in this paper can, and are used as subroutines (gadgets) within other approximation algorithms.