Neural Control Variates

Thomas Müller, Fabrice Rousselle, Alexander Keller, Jan Novák

Transaction on Graphics (Proceedings of SIGGRAPH Asia 2020), vol. 39, no. 6

When applied to light-transport simulation, our neural control-variate algorithm (a) learns an approximation 𝐺 of the scattered light field and corrects the approximation error by estimating the difference between the original integrand 𝑓 and the corresponding learned control variate 𝑔. This is enabled by our construction that couples 𝑔 and 𝐺 such that 𝑔 always exactly integrates to 𝐺. To further reduce noise, we importance sample the absolute difference |𝑓 − 𝑔| using a learned probability density function (PDF) 𝑝 (b). We also provide a heuristic (c) to terminate paths without estimating the difference. This reduces the mean path length and removes most of the remaining noise. On the right (d), we compare the error of rendering the Veach Door scene using an improved variant (NIS++) of neural importance sampling [Müller et al. 2019] to our neural control variates (NCV) with and without our path termination heuristic.

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.

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bibtex

@article{mueller2020ncv,
	author = {Thomas M\"{u}ller and Fabrice Rousselle and Alexander Keller and Jan Nov\'{a}k},
	title = {Neural Control Variates},
	journal = {ACM Trans. Graph.},
	issue_date = {December 2020},
	volume = {39},
	number = {6},
	month = dec,
	year = {2020},
	articleno = {243},
	numpages = {19},
	url = {http://doi.acm.org/10.1145/3414685.3417804},
	doi = {10.1145/3414685.3417804},
	publisher = {ACM},
	address = {New York, NY, USA},
}