REBAR: Low-variance, unbiased gradient estimates for discrete latent variable models
Learning in models with discrete latent variables is challenging due to high variance gradient estimators. Generally, approaches have relied on control variates to reduce the variance of the REINFORCE estimator. Recent work (Jang et al., 2016;Maddison et al., 2016) has taken a different approach, introducing a continuous relaxation of discrete variables to produce low-variance, but biased, gradient estimates. In this work, we combine the two approaches through a novel control variate that produces low-variance, unbiased gradient estimates. Then, we introduce a modification to the continuous relaxation and show that the tightness of the relaxation can be adapted online, removing it as a hyperparameter. We show state-of-the-art variance reduction on several benchmark generative modeling tasks, generally leading to faster convergence to a better final log-likelihood.
to significantly improve its effectiveness. We call this the REBAR gradient estimator, because it combines REINFORCE gradients with gradients of the Concrete relaxation. Next, we show that a modification to the Concrete relaxation connects REBAR to MuProp in the high temperature limit. Finally, because REBAR is unbiased for all temperatures, we show that the temperature can be optimized online to reduce variance further and relieve the burden of setting an additional hyperparameter.
In our experiments, we illustrate the potential problems inherent with biased gradient estimators on a toy problem. Then, we use REBAR to train generative sigmoid belief networks (SBNs) on the MNIST and Omniglot datasets and to train conditional generative models on MNIST. Across tasks, we show that REBAR has state-of-the-art variance reduction which translates to faster convergence and better final log-likelihoods. Although we focus on binary variables for simplicity, this work is equally applicable to categorical variables (Appendix C).
This paper introduces a control variate technique to reduce the variance of the REINFORCE gradient estimator for discrete latent variables. The method, called REBAR, is inspired by the Gumble-softmax/Concrete relaxations; however, in contrast to those, REBAR provides an unbiased gradient estimator. The paper shows a connection between REBAR and MuProp. The variance of the REBAR estimator is compared to state-of-the-art methods on sigmoid belief networks. The paper focuses on binary discrete latent variables.
Overall, I found this is an interesting paper that addresses a relevant problem; namely, non-expensive low-variance gradient estimators for discrete latent variables. The writing quality is good, although there are some parts that weren't clear to me (see comments below). The connections to MuProp and Gumble-softmax/Concrete are clear.
Please find below a list with detailed comments and concerns:
- The paper states several times that p(z)=p(z|b)p(b). This is confusing, as p(z|b)p(b) should be the joint p(z,b). I think that the point is that the joint can be alternatively written as p(z,b)=p(b|z)p(z), where the first term is an indicator function, which takes value 1 if b=H(z) and zero otherwise, and that it motivates dropping this term. But being rigorous, the indicator function should be kept. So p(b|z)p(z)=p(z|b)p(b), and when b=H(z), then p(z)=p(z|b)p(b). I don't think the derivations in the paper are wrong, but this issue was confusing to me and should be clarified.
- It is not clear to me how the key equation in the paper was obtained (the unnumbered equation between lines 98-99). The paper just reads "Putting the terms together, we arrive at", but this wasn't clear to me. The paper would definitely be improved by adding these details in the main text or in the Appendix.
- In the same equation, the expectation w.r.t. p(u,v), where u, v ~ Uniform(0,1) is also misleading, because they're not independent. As Appendix 7.5 reads, "a choice of u will determine a corresponding choice of v which produces the same z", i.e., p(u,v)=p(u)p(v|u), with p(v|u) being deterministic. In other words, I don't see how "using this pair (u,v) as the random numbers" is a choice (as the paper suggests) rather than a mathematical consequence of the procedure.
- The paper states that REBAR is equally applicable to the non-binary case (lines 44-45). I think the paper can be significantly improved by including the mathematical details in the Appendix to make this explicit.
- In the experimental section, all figures show "steps" (of the variational procedure) in the x-axis. This should be replaced with wall-clock time to get a sense of the computational complexity of each approach. 
- In the experiments, it would be interesting to see a comparison with "Local expectation gradients" [Titsias & Lazaro-Gredilla].
- In the experimental section, do the authors have any intuition on why MuProp excels in the structured prediction task for Omniglot? (Figs. 6 and 7)
- The procedure to optimize temperature (Sec 3.2) seems related to the procedure to optimize temperature in "Overdispersed black-box VI" [Ruiz et al.]; if that's the case, the connection can be pointed out.
- The multilayer section (Sec 3.3) seems a Rao-Blackwellization procedure (see, e.g., [Ranganath et al.]). If that's the case, the authors should mention that.
- In line 108, there seems to be a missing 1/lambda term.
- In lines 143-144 and 193, there are missing parenthesis in the citations.
- Consider renaming Appendix 7 as Appendix 1.