Improved Gradient-Based Optimization Over Discrete Distributions

In many applications we seek to maximize an expectation with respect to a distribution over discrete variables. Estimating gradients of such objectives with respect to the distribution parameters is a challenging problem. We analyze existing solutions including finite-difference (FD) estimators and continuous relaxation (CR) estimators in terms of bias and variance. We show that the commonly used Gumbel-Softmax estimator is biased and propose a simple method to reduce it. We also derive a simpler piece-wise linear continuous relaxation that also possesses reduced bias. We demonstrate empirically that reduced bias leads to a better performance in variational inference and on binary optimization tasks

Undirected Graphical Models as Approximate Posteriors

The representation of the approximate posterior is a critical aspect of effective variational autoencoders (VAEs). Poor choices for the approximate posterior have a detrimental impact on the generative performance of VAEs due to the mismatch with the true posterior. We extend the class of posterior models that may be learned by using undirected graphical models. We develop an efficient method to train undirected approximate posteriors by showing that the gradient of the training objective with respect to the parameters of the undirected posterior can be computed by backpropagation through Markov chain Monte Carlo updates. We apply these gradient estimators for training discrete VAEs with Boltzmann machines as approximate posteriors and demonstrate that undirected models outperform previous results obtained using directed graphical models. Our implementation is available at https://github.com/QuadrantAI/dvaess .Comment: Accepted to ICML 202

ARSM: Augment-REINFORCE-Swap-Merge Estimator for Gradient Backpropagation Through Categorical Variables

To address the challenge of backpropagating the gradient through categorical variables, we propose the augment-REINFORCE-swap-merge (ARSM) gradient estimator that is unbiased and has low variance. ARSM first uses variable augmentation, REINFORCE, and Rao-Blackwellization to re-express the gradient as an expectation under the Dirichlet distribution, then uses variable swapping to construct differently expressed but equivalent expectations, and finally shares common random numbers between these expectations to achieve significant variance reduction. Experimental results show ARSM closely resembles the performance of the true gradient for optimization in univariate settings; outperforms existing estimators by a large margin when applied to categorical variational auto-encoders; and provides a "try-and-see self-critic" variance reduction method for discrete-action policy gradient, which removes the need of estimating baselines by generating a random number of pseudo actions and estimating their action-value functions.Comment: Published in ICML 2019. We have updated Section 4.2 and the Appendix to reflect the improvements brought by fixing some bugs hidden in our original code. Please find the Errata in the authors' websites and check the updated code in Githu