Demystifying Orthogonal Monte Carlo and Beyond

TL;DR

This paper provides new theoretical insights into Orthogonal Monte Carlo (OMC), introduces an improved algorithm called Near-Orthogonal Monte Carlo (NOMC), and demonstrates NOMC's superior performance across various machine learning tasks.

Contribution

It offers a theoretical analysis of OMC using negatively dependent variables and proposes NOMC, a novel extension that outperforms OMC in multiple applications.

Findings

NOMC consistently outperforms OMC in kernel methods.

New concentration results for OMC using negative dependence theory.

NOMC leverages number theory and particle algorithms for improved performance.

Abstract

Orthogonal Monte Carlo (OMC) is a very effective sampling algorithm imposing structural geometric conditions (orthogonality) on samples for variance reduction. Due to its simplicity and superior performance as compared to its Quasi Monte Carlo counterparts, OMC is used in a wide spectrum of challenging machine learning applications ranging from scalable kernel methods to predictive recurrent neural networks, generative models and reinforcement learning. However theoretical understanding of the method remains very limited. In this paper we shed new light on the theoretical principles behind OMC, applying theory of negatively dependent random variables to obtain several new concentration results. We also propose a novel extensions of the method leveraging number theory techniques and particle algorithms, called Near-Orthogonal Monte Carlo (NOMC). We show that NOMC is the first algorithm consistently outperforming OMC in applications ranging from kernel methods to approximating distances in probabilistic metric spaces.