Stochastic variational inference with an amortized inference model and the reparameterization trick has become a widely-used algorithm for learning latent variable models. Increasing the flexibility of approximate posterior distributions while maintaining computational tractability is one of the core problems in stochastic variational inference. Two families of approaches proposed to address the problem are flow-based and multisample-based approaches such as importance weighted auto-encoders (IWAE). We introduce a new learning algorithm, the annealed importance weighted auto-encoder (AIWAE), for learning latent variable models. The proposed AIWAE combines multisample-based and flow-based approaches with the annealed importance sampling and its memory cost stays constant when the depth of flows increases. The flow constructed using an annealing process in AIWAE facilitates the exploration of the latent space when the posterior distribution has multiple modes. Through computational experiments, we show that, compared to models trained using the IWAE, AIWAE-trained models are better density models, have more complex posterior distributions and use more latent space representation capacity.
In addition to the IWAE and AIWAE, several approaches have been proposed to utilize multiple samples with importance sampling to either obtain a better objective function or to better estimate gradients of objective functions. For learning feedforward neural networks with stochastic binary hidden units, an importance sampling based estimator using multiple samples was proposed in @cite_22 to estimate the expectation used in the generalized Expectation-Maximization algorithm @cite_18 . A similar estimator was derived in @cite_1 by constructing a variational distribution of hidden variables used in ELBO using multiple samples and importance sampling. Another approach @cite_11 extended the multisample approach in IWAE to discrete latent variables and developed an unbiased gradient estimator for importance-sampled objectives. Methods in @cite_22 @cite_1 @cite_11 apply for models with discrete latent variables, whereas the IWAE only applies for models with continuous latent variables. Although our current extension of IWAE to AIWAE using AIS also only applies for continuous latent variables, the idea of replacing importance sampling with AIS could also be useful for improving methods in @cite_22 @cite_1 @cite_11 that work for discrete latent variables.
@cite_18 @cite_1 @cite_22 @cite_11
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The EM algorithm performs maximum likelihood estimation for data in which some variables are unobserved. We present a function that resembles negative free energy and show that the M step maximizes this function with respect to the model parameters and the E step maximizes it with respect to the distribution over the unobserved variables. From this perspective, it is easy to justify an incremental variant of the EM algorithm in which the distribution for only one of the unobserved variables is recalculated in each E step. This variant is shown empirically to give faster convergence in a mixture estimation problem. A variant of the algorithm that exploits sparse conditional distributions is also described, and a wide range of other variant algorithms are also seen to be possible. Stochastic binary hidden units in a multi-layer perceptron (MLP) network give at least three potential benefits when compared to deterministic MLP networks. (1) They allow to learn one-to-many type of mappings. (2) They can be used in structured prediction problems, where modeling the internal structure of the output is important. (3) Stochasticity has been shown to be an excellent regularizer, which makes generalization performance potentially better in general. However, training stochastic networks is considerably more difficult. We study training using M samples of hidden activations per input. We show that the case M=1 leads to a fundamentally different behavior where the network tries to avoid stochasticity. We propose two new estimators for the training gradient and propose benchmark tests for comparing training algorithms. Our experiments confirm that training stochastic networks is difficult and show that the proposed two estimators perform favorably among all the five known estimators. Multilayer perceptrons (MLPs) or neural networks are popular models used for nonlinear regression and classification tasks. As regressors, MLPs model the conditional distribution of the predictor variables Y given the input variables X. However, this predictive distribution is assumed to be unimodal (e.g. Gaussian). For tasks involving structured prediction, the conditional distribution should be multi-modal, resulting in one-to-many mappings. By using stochastic hidden variables rather than deterministic ones, Sigmoid Belief Nets (SBNs) can induce a rich multimodal distribution in the output space. However, previously proposed learning algorithms for SBNs are not efficient and unsuitable for modeling real-valued data. In this paper, we propose a stochastic feedforward network with hidden layers composed of both deterministic and stochastic variables. A new Generalized EM training procedure using importance sampling allows us to efficiently learn complicated conditional distributions. Our model achieves superior performance on synthetic and facial expressions datasets compared to conditional Restricted Boltzmann Machines and Mixture Density Networks. In addition, the latent features of our model improves classification and can learn to generate colorful textures of objects. Recent progress in deep latent variable models has largely been driven by the development of flexible and scalable variational inference methods. Variational training of this type involves maximizing a lower bound on the log-likelihood, using samples from the variational posterior to compute the required gradients. Recently, (2016) have derived a tighter lower bound using a multi-sample importance sampling estimate of the likelihood and showed that optimizing it yields models that use more of their capacity and achieve higher likelihoods. This development showed the importance of such multi-sample objectives and explained the success of several related approaches. We extend the multi-sample approach to discrete latent variables and analyze the difficulty encountered when estimating the gradients involved. We then develop the first unbiased gradient estimator designed for importance-sampled objectives and evaluate it at training generative and structured output prediction models. The resulting estimator, which is based on low-variance per-sample learning signals, is both simpler and more effective than the NVIL estimator proposed for the single-sample variational objective, and is competitive with the currently used biased estimators.