Neural Control Variates

TL;DR

This paper introduces neural control variates (NCV), a novel method using neural networks to achieve unbiased variance reduction in Monte Carlo integration, especially effective in light transport simulation.

Contribution

The paper presents a new neural network-based approach for variance reduction in Monte Carlo methods, including a variance-minimizing loss function and a neural importance sampler.

Findings

Neural control variates match state-of-the-art unbiased methods in light transport.

The approach reduces noise significantly with negligible bias.

The method enables high-order bounces without error correction.

Abstract

We propose neural control variates (NCV) for unbiased variance reduction in parametric Monte Carlo integration. So far, the core challenge of applying the method of control variates has been finding a good approximation of the integrand that is cheap to integrate. We show that a set of neural networks can face that challenge: a normalizing flow that approximates the shape of the integrand and another neural network that infers the solution of the integral equation. We also propose to leverage a neural importance sampler to estimate the difference between the original integrand and the learned control variate. To optimize the resulting parametric estimator, we derive a theoretically optimal, variance-minimizing loss function, and propose an alternative, composite loss for stable online training in practice. When applied to light transport simulation, neural control variates are capable of matching the state-of-the-art performance of other unbiased approaches, while providing means to develop more performant, practical solutions. Specifically, we show that the learned light-field approximation is of sufficient quality for high-order bounces, allowing us to omit the error correction and thereby dramatically reduce the noise at the cost of negligible visible bias.