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Title: Learning to Infer

Paper content:
Inference models, which replace an optimization-based inference procedure with a learned model, have been fundamental in advancing Bayesian deep learning, the most notable example being variational auto-encoders (VAEs).
In this paper, we propose iterative inference models, which learn how to optimize a variational lower bound through repeatedly encoding gradients.
Our approach generalizes VAEs under certain conditions, and by viewing VAEs in the context of iterative inference, we provide further insight into several recent empirical findings.
We demonstrate the inference optimization capabilities of iterative inference models, explore unique aspects of these models, and show that they outperform standard inference models on typical benchmark data sets.
Generative models present the possibility of learning structure from data in unsupervised or semisupervised settings, thereby facilitating more flexible systems to learn and perform tasks in computer vision, robotics, and other application domains with limited human involvement.
Latent variable models, a class of generative models, are particularly well-suited to learning hidden structure.
They frame the process of data generation as a mapping from a set of latent variables underlying the data.
When this mapping is parameterized by a deep neural network, the model can learn complex, non-linear relationships, such as object identities (Higgins et al. (2016) ) and dynamics (Xue et al. (2016) ; Karl et al. (2017) ).
However, performing exact posterior inference in these models is computationally intractable, necessitating the use of approximate inference methods.Variational inference (Hinton & Van Camp (1993) ; Jordan et al. (1998) ) is a scalable approximate inference method, transforming inference into a non-convex optimization problem.
Using a set of approximate posterior distributions, e.g. Gaussians, variational inference attempts to find the distribution that most closely matches the true posterior.
This matching is accomplished by maximizing a lower bound on the marginal log-likelihood, or model evidence, which can also be used to learn the model parameters.
The ensuing expectation-maximization procedure alternates between optimizing the approximate posteriors and model parameters (Dempster et al. (1977) ; Neal & Hinton (1998) ; Hoffman et al. (2013) ).
Amortized inference (Gershman & Goodman (2014) ) avoids exactly computing optimized approximate posterior distributions for each data example, instead learning a separate inference model to perform this task.
Taking the data example as input, this model outputs an estimate of the corresponding approximate posterior.
When the generative and inference models are parameterized with neural networks, the resulting set-up is referred to as a variational auto-encoder (VAE) (Kingma & Welling (2014) ; Rezende et al. (2014) ).We
introduce a new class of inference models, referred to as iterative inference models, inspired by recent work in learning to learn (Andrychowicz et al. (2016) ). Rather
than directly mapping the data to the approximate posterior, these models learn how to iteratively estimate the approximate posterior by repeatedly encoding the corresponding gradients, i.e. learning to infer. With inference
computation distributed over multiple iterations, we conjecture that this model set-up should provide improved inference estimates over standard inference models given sufficient model capacity. Our work is presented
as follows: Section 2 contains background on latent variable models, variational inference, and inference models; Section 3 motivates and introduces iterative inference models; Section 4 presents this approach for latent Gaussian models, showing that a particular form of iterative inference models reduces to standard inference models under mild assumptions; Section 5 contains empirical results; and Section 6 concludes our work.

assistant: text: We propose a new class of inference models that iteratively encode gradients to estimate approximate posterior distributions.