When optimizing against the mean loss over a distribution of predictions in the context of a regression task, then even if there is a distribution of targets the optimal prediction distribution is always a delta function at a single value. Methods of constructing generative models need to overcome this tendency. We consider a simple method of summarizing the prediction error, such that the optimal strategy corresponds to outputting a distribution of predictions with a support that matches the support of the distribution of targets --- optimizing against the minimal value of the loss given a set of samples from the prediction distribution, rather than the mean. We show that models trained against this loss learn to capture the support of the target distribution and, when combined with an auxiliary classifier-like prediction task, can be projected via rejection sampling to reproduce the full distribution of targets. The resulting method works well compared to other generative modeling approaches particularly in low dimensional spaces with highly non-trivial distributions, due to mode collapse solutions being globally suboptimal with respect to the extreme value loss. However, the method is less suited to high-dimensional spaces such as images due to the scaling of the number of samples needed in order to accurately estimate the extreme value loss when the dimension of the data manifold becomes large.
: One method for generating samples from a high-dimensional distribution is to factorize the joint distribution into a product of one-dimensional conditional probabilities: @math . These one-dimensional distributions can each be learned separately by discretizing the possibilities and minimizing the categorical cross-entropy or KL divergence. Alternately, if the variables @math have some structural relationship, such as words in a sentence or pixels in an image, then weight sharing and causal masking can be used to learn a single model which is able to generate all such distributions. Sampling from the distribution is done by sampling from each one-dimensional distribution in turn, conditioning on the previous samples. Alternately, methods such as beam-search can be used to find maximum likelihood outcomes or otherwise adjust the 'temperature' of the generator. Methods which use this approach include CharRNN @cite_18 , PixelRNN @cite_4 , PixelCNN @cite_25 @cite_28 , WaveNet @cite_3 , and Transformer networks @cite_9 .
@cite_18 @cite_4 @cite_28 @cite_9 @cite_3 @cite_25
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Recurrent Neural Networks (RNNs) are very powerful sequence models that do not enjoy widespread use because it is extremely difficult to train them properly. Fortunately, recent advances in Hessian-free optimization have been able to overcome the difficulties associated with training RNNs, making it possible to apply them successfully to challenging sequence problems. In this paper we demonstrate the power of RNNs trained with the new Hessian-Free optimizer (HF) by applying them to character-level language modeling tasks. The standard RNN architecture, while effective, is not ideally suited for such tasks, so we introduce a new RNN variant that uses multiplicative (or "gated") connections which allow the current input character to determine the transition matrix from one hidden state vector to the next. After training the multiplicative RNN with the HF optimizer for five days on 8 high-end Graphics Processing Units, we were able to surpass the performance of the best previous single method for character-level language modeling – a hierarchical non-parametric sequence model. To our knowledge this represents the largest recurrent neural network application to date. Modeling the distribution of natural images is a landmark problem in unsupervised learning. This task requires an image model that is at once expressive, tractable and scalable. We present a deep neural network that sequentially predicts the pixels in an image along the two spatial dimensions. Our method models the discrete probability of the raw pixel values and encodes the complete set of dependencies in the image. Architectural novelties include fast two-dimensional recurrent layers and an effective use of residual connections in deep recurrent networks. We achieve log-likelihood scores on natural images that are considerably better than the previous state of the art. Our main results also provide benchmarks on the diverse ImageNet dataset. Samples generated from the model appear crisp, varied and globally coherent. The dominant sequence transduction models are based on complex recurrent or convolutional neural networks in an encoder-decoder configuration. The best performing models also connect the encoder and decoder through an attention mechanism. We propose a new simple network architecture, the Transformer, based solely on attention mechanisms, dispensing with recurrence and convolutions entirely. Experiments on two machine translation tasks show these models to be superior in quality while being more parallelizable and requiring significantly less time to train. Our model achieves 28.4 BLEU on the WMT 2014 English-to-German translation task, improving over the existing best results, including ensembles by over 2 BLEU. On the WMT 2014 English-to-French translation task, our model establishes a new single-model state-of-the-art BLEU score of 41.8 after training for 3.5 days on eight GPUs, a small fraction of the training costs of the best models from the literature. We show that the Transformer generalizes well to other tasks by applying it successfully to English constituency parsing both with large and limited training data. This work explores conditional image generation with a new image density model based on the PixelCNN architecture. The model can be conditioned on any vector, including descriptive labels or tags, or latent embeddings created by other networks. When conditioned on class labels from the ImageNet database, the model is able to generate diverse, realistic scenes representing distinct animals, objects, landscapes and structures. When conditioned on an embedding produced by a convolutional network given a single image of an unseen face, it generates a variety of new portraits of the same person with different facial expressions, poses and lighting conditions. We also show that conditional PixelCNN can serve as a powerful decoder in an image autoencoder. Additionally, the gated convolutional layers in the proposed model improve the log-likelihood of PixelCNN to match the state-of-the-art performance of PixelRNN on ImageNet, with greatly reduced computational cost.