Reparameterization trick for discrete variables
This addresses a bottleneck in variational learning for discrete latent variable models, such as sigmoid belief networks, by enabling more efficient training, though it is an incremental improvement over existing gradient estimation techniques.
The paper tackles the problem of low-variance gradient estimation for discrete variables in directed graphical models, where the reparameterization trick was previously not applicable due to discontinuity. It introduces a new reparameterization trick that bypasses this by marginalizing out the variable, resulting in a theoretically guaranteed variance reduction compared to the likelihood-ratio method with optimal baseline.
Low-variance gradient estimation is crucial for learning directed graphical models parameterized by neural networks, where the reparameterization trick is widely used for those with continuous variables. While this technique gives low-variance gradient estimates, it has not been directly applicable to discrete variables, the sampling of which inherently requires discontinuous operations. We argue that the discontinuity can be bypassed by marginalizing out the variable of interest, which results in a new reparameterization trick for discrete variables. This reparameterization greatly reduces the variance, which is understood by regarding the method as an application of common random numbers to the estimation. The resulting estimator is theoretically guaranteed to have a variance not larger than that of the likelihood-ratio method with the optimal input-dependent baseline. We give empirical results for variational learning of sigmoid belief networks.