Smooth Convex Optimization using Sub-Zeroth-Order Oracles

TL;DR

This paper introduces new algorithms for convex optimization using sub-zeroth-order oracles, achieving polynomial sample complexity and the first known polynomial convergence rate for comparator oracles.

Contribution

It presents the first polynomial convergence rate algorithm for comparator oracles and analyzes sample complexity for various sub-zeroth-order oracles in convex optimization.

Findings

Sample complexity is polynomial in relevant parameters.

First polynomial convergence rate for comparator oracle-based optimization.

Regret bound of (n^{3.75} T^{0.75}) for noisy-value oracle.

Abstract

We consider the problem of minimizing a smooth, Lipschitz, convex function over a compact, convex set using sub-zeroth-order oracles: an oracle that outputs the sign of the directional derivative for a given point and a given direction, an oracle that compares the function values for a given pair of points, and an oracle that outputs a noisy function value for a given point. We show that the sample complexity of optimization using these oracles is polynomial in the relevant parameters. The optimization algorithm that we provide for the comparator oracle is the first algorithm with a known rate of convergence that is polynomial in the number of dimensions. We also give an algorithm for the noisy-value oracle that incurs a regret of $\tilde{\mathcal{O}}(n^{3.75} T^{0.75})$ (ignoring the other factors and logarithmic dependencies) where $n$ is the number of dimensions and $T$ is the number of queries.