Filtering Variational Objectives
When used as a surrogate objective for maximum likelihood estimation in latent variable models, the evidence lower bound (ELBO) produces state-of-the-art results. Inspired by this, we consider the extension of the ELBO to a family of lower bounds defined by a particle filter's estimator of the marginal likelihood, the filtering variational objectives (FIVOs). FIVOs take the same arguments as the ELBO, but can exploit a model's sequential structure to form tighter bounds. We present results that relate the tightness of FIVO's bound to the variance of the particle filter's estimator by considering the generic case of bounds defined as log-transformed likelihood estimators. Experimentally, we show that training with FIVO results in substantial improvements over training the same model architecture with the ELBO on sequential data.
This paper introduces Monte Carlo objective to improve the scaling of the IWAE and provides a tighter lower bound to the log likelihood than traditional ELBO. Various theoretical properties of Monte Carlo objectives are discussed. An example of Monte Carlo objective is provided to use particle filters for modeling sequential data. The paper is generally well written and I enjoyed reading it.
* For the FIVO algorithm, it'd be more self-contained if including a brief description of the resampling criteria or its exact procedure used in the experiment, so that the reference doesn't need to be consulted.
* Although not required, it'd be helpful to provide extra examples of Monte Carlo objectives other than applications of sequential data as FIVOs.
* I feel it's fairer to stay with a batch of 4 instead of 4N for ELBO to reduce the impact of different batch configurations, even if the computational budget does not match up (line 259). The authors could also consider reporting the running time of each method, and normalize it somehow when interpreting the results. This could also give readers some idea about the overhead of resampling.
* I'm a bit worried about the use of the biased gradients to train FIVO. Is it possible that part of the superior performance of FIVO over IWAE somehow benefits from training using the incorrect gradients? It'd be useful to also report the performance of FIVO trained with the unbiased gradients so that the readers will have the idea about the impact of the large variance coming from the resampling term.
* It's not clear to me why checking on the KL divergence between the inference model q(z|x) to the prior p(z) addresses the need to make use of the uncertainty (line 279 to 283). What's the connection between "the degree to which each model uses the latent states" and the KL divergence? How to interpret ELBO's KL collapses during training? Doesn't the inference model collapses onto the prior suggest that it carries less information about the data?
* Typo: it should be z ~ q(z|x) in line 111-112.