Fourier analysis is gaining popularity in image synthesis as a tool for the analysis of error in Monte Carlo (MC) integration. Still, existing tools are only able to analyse convergence under simplifying assumptions (such as randomized shifts) which are not applied in practice during rendering. We reformulate the expressions for bias and variance of sampling‐based integrators to unify non‐uniform sample distributions [importance sampling (IS)] as well as correlations between samples while respecting finite sampling domains. Our unified formulation hints at fundamental limitations of Fourier‐based tools in performing variance analysis for MC integration. At the same time, it reveals that, when combined with correlated sampling, IS can impact convergence rate by introducing or inhibiting discontinuities in the integrand. We demonstrate that the convergence of multiple importance sampling (MIS) is determined by the strategy which converges slowest and propose several simple approaches to overcome this limitation. We show that smoothing light boundaries (as commonly done in production to reduce variance) can improve (M)IS convergence (at a cost of introducing a small amount of bias) since it removes
In reinforcement learning, importance sampling is a widely used method for evaluating an expectation under the distribution of data of one policy when the data has in fact been generated by a different policy. Importance sampling requires computing the likelihood ratio between the action probabilities of a target policy and those of the data-producing behavior policy. In this article, we study importance sampling where the behavior policy action probabilities are replaced by their maximum likelihood estimate of these probabilities under the observed data. We show this general technique reduces variance due to sampling error in Monte Carlo style estimators. We introduce two novel estimators that use this technique to estimate expected values that arise in the RL literature. We find that these general estimators reduce the variance of Monte Carlo sampling methods, leading to faster learning for policy gradient algorithms and more accurate off-policy policy evaluation. We also provide theoretical analysis showing that our new estimators are consistent and have asymptotically lower variance than Monte Carlo estimators.
more » « less- NSF-PAR ID:
- 10226772
- Publisher / Repository:
- Springer Science + Business Media
- Date Published:
- Journal Name:
- Machine Learning
- Volume:
- 110
- Issue:
- 6
- ISSN:
- 0885-6125
- Page Range / eLocation ID:
- p. 1267-1317
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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