  We provide theoretical and empirical evidence that using tighter evidence
lower bounds (ELBOs) can be detrimental to the process of learning an inference
network by reducing the signal-to-noise ratio of the gradient estimator. Our
results call into question common implicit assumptions that tighter ELBOs are
better variational objectives for simultaneous model learning and inference
amortization schemes. Based on our insights, we introduce three new algorithms:
the partially importance weighted auto-encoder (PIWAE), the multiply importance
weighted auto-encoder (MIWAE), and the combination importance weighted
auto-encoder (CIWAE), each of which includes the standard importance weighted
auto-encoder (IWAE) as a special case. We show that each can deliver
improvements over IWAE, even when performance is measured by the IWAE target
itself. Furthermore, our results suggest that PIWAE may be able to deliver
simultaneous improvements in the training of both the inference and generative
networks.
