Optimal Variance Control of the Score-Function Gradient Estimator for Importance-Weighted Bounds

This paper introduces novel results for the score function gradient estimator of the importance weighted variational bound (IWAE). We prove that in the limit of large K (number of importance samples) one can choose the control variate such that the Signal-to-Noise ratio (SNR) of the estimator grows as √ K. This is in contrast to the standard pathwise gradient estimator where the SNR decreases as 1/ √ K. Based on our theoretical findings we develop a novel control variate that extends on VIMCO. Empirically, for the training of both continuous and discrete generative models, the proposed method yields superior variance reduction, resulting in an SNR for IWAE that increases with K without relying on the reparameterization trick. The novel estimator is competitive with state-of-the-art reparameterization-free gradient estimators such as Reweighted Wake-Sleep (RWS) and the thermodynamic variational objective (TVO) when training generative models. Recently, variational objectives tighter than the traditional evidence lower bound (ELBO) have been proposed [21, 22] . In importance weighted autoencoders (IWAE) [22] the tighter bound comes with the price of a K-fold increase in the required number of samples from the inference network. Despite yielding a tighter bound, using more samples can be detrimental to the learning of the inference model [23] . In fact, the Signal-to-Noise ratio (the ratio of the expected gradient to its standard deviation) of the pathwise estimator has been shown to decrease at a rate O(K -1/2 ) [23] . Although this can be improved to O(K 1/2 ) by exploiting properties of the gradient to cancel high-variance 34th Conference on Neural Information Processing Systems (NeurIPS 2020),