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Title: Doubly Reparameterized Gradient Estimators for Monte Carlo Objectives

Paper content:
Deep latent variable models have become a popular model choice due to the scalable learning algorithms introduced by (Kingma & Welling 2013, Rezende et al. 2014).
These approaches maximize a variational lower bound on the intractable log likelihood of the observed data.
Burda et al. (2015) introduced a multi-sample variational bound, IWAE, that is at least as tight as the standard variational lower bound and becomes increasingly tight as the number of samples increases.
Counterintuitively, the typical inference network gradient estimator for the IWAE bound performs poorly as the number of samples increases (Rainforth et al. 2018, Le et al. 2018).
Roeder et a.
(2017) propose an improved gradient estimator, however, are unable to show it is unbiased.
We show that it is in fact biased and that the bias can be estimated efficiently with a second application of the reparameterization trick.
The doubly reparameterized gradient (DReG) estimator does not suffer as the number of samples increases, resolving the previously raised issues.
The same idea can be used to improve many recently introduced training techniques for latent variable models.
In particular, we show that this estimator reduces the variance of the IWAE gradient, the reweighted wake-sleep update (RWS) (Bornschein & Bengio 2014), and the jackknife variational inference (JVI) gradient (Nowozin 2018).
Finally, we show that this computationally efficient, drop-in estimator translates to improved performance for all three objectives on several modeling tasks.
Following the influential work by BID20 BID30 , deep generative models with latent variables have been widely used to model data such as natural images BID29 BID14 , speech and music time-series BID8 BID11 BID22 , and video BID1 BID15 BID9 .
The power of these models lies in combining learned nonlinear function approximators with a principled probabilistic approach, resulting in expressive models that can capture complex distributions.
Unfortunately, the nonlinearities that empower these model also make marginalizing the latent variables intractable, rendering direct maximum likelihood training inapplicable.
Instead of directly maximizing the marginal likelihood, a common approach is to maximize a tractable lower bound on the likelihood such as the variational evidence lower bound (ELBO) BID19 BID3 .
The tightness of the bound is determined by the expressiveness of the variational family.
For tractability, a factorized variational family is commonly used, which can cause the learned model to be overly simplistic.
BID5 introduced a multi-sample bound, IWAE, that is at least as tight as the ELBO and becomes increasingly tight as the number of samples increases.
Counterintuitively, although the bound is tighter, BID28 theoretically and empirically showed that the standard inference network gradient estimator for the IWAE bound performs poorly as the number of samples increases due to a diminishing signal-to-noise ratio (SNR).
This motivates the search for novel gradient estimators.
BID31 proposed a lower-variance estimator of the gradient of the IWAE bound.
They speculated that their estimator was unbiased, however, were unable to prove the claim.
We show that it is in fact biased, but that it is possible to construct an unbiased estimator with a second application of the reparameterization trick which we call the IWAE doubly reparameterized gradient (DReG) estimator.
Our estimator is an unbiased, computationally efficient drop-in replacement, and does not suffer as the number of samples increases, resolving the counterintuitive behavior from previous work BID28 .
Furthermore, our insight is applicable to alternative multisample training techniques for latent variable models: reweighted wake-sleep (RWS) BID4 and jackknife variational inference (JVI) BID27 .In
this work, we derive DReG estimators for IWAE, RWS, and JVI and demonstrate improved scaling with the number of samples on a simple example. Then
, we evaluate DReG estimators on MNIST generative modeling, Omniglot generative modeling, and MNIST structured prediction tasks. In
all cases, we demonstrate substantial unbiased variance reduction, which translates to improved performance over the original estimators.
In this work, we introduce doubly reparameterized estimators for the updates in IWAE, RWS, and JVI.
We demonstrate that across tasks they provide unbiased variance reduction, which leads to improved performance.
Furthermore, DReG estimators have the same computational cost as the original estimators.
As a result, we recommend that DReG estimators be used instead of the typical gradient estimators.Variational Sequential Monte Carlo BID24 BID26 and Neural Adapative Sequential Monte Carlo BID13 extend IWAE and RWS to sequential latent variable models, respectively.
It would be interesting to develop DReG estimators for these approaches as well.We found that a convex combination of IWAE-DReG and RWS-DReG performed best, however, the weighting was task dependent.
In future work, we intend to apply ideas from BID2 to automatically adapt the weighting based on the data.Finally, the form of the IWAE-DReG estimator (Eq. 7) is surprisingly simple and suggests that there may be a more direct derivation that is applicable to general MCOs.

assistant: text: Doubly reparameterized gradient estimators provide unbiased variance reduction which leads to improved performance.