Sample Efficient Graph-Based Optimization with Noisy Observations

Abstract

We study sample complexity of optimizing "hill-climbing friendly" functions defined on a graph under noisy observations. We define a notion of convexity, and we show that a variant of best-arm identification can find a near-optimal solution after a small number of queries that is independent of the size of the graph. For functions that have local minima and are nearly convex, we show a sample complexity for the classical simulated annealing under noisy observations. We show effectiveness of the greedy algorithm with restarts and the simulated annealing on problems of graph-based nearest neighbor classification as well as a web document re-ranking application.

The first version of this paper appeared in AISTATS 2019. Thank to community feedback, some typos and a minor issue have been identified. Specifically, on page 4, column 2, line 18, the statement ${\Delta }_{1,s}\ge (1+m{)}^{S-1-s}{\Delta }_1$ is not valid, and in the proof of Theorem 2, "By Lemma 1" should be "By Definition 2". These problems are fixed in this updated version published here on arxiv