Variance reduction properties of the reparameterization trick
This provides theoretical insight into a widely used technique in machine learning, though it is incremental as it builds on existing empirical evidence.
The paper investigates why the reparameterization trick is effective in variational inference, showing under idealized assumptions that it reduces marginal variances compared to the score function method, and applies this analysis to real-world examples.
The reparameterization trick is widely used in variational inference as it yields more accurate estimates of the gradient of the variational objective than alternative approaches such as the score function method. Although there is overwhelming empirical evidence in the literature showing its success, there is relatively little research exploring why the reparameterization trick is so effective. We explore this under the idealized assumptions that the variational approximation is a mean-field Gaussian density and that the log of the joint density of the model parameters and the data is a quadratic function that depends on the variational mean. From this, we show that the marginal variances of the reparameterization gradient estimator are smaller than those of the score function gradient estimator. We apply the result of our idealized analysis to real-world examples.