This work presents a new algorithm for training recurrent neural networks (although ideas are applicable to feedforward networks as well). The algorithm is derived from a theory in nonconvex optimization related to the diffusion equation. The contributions made in this work are two fold. First, we show how some seemingly disconnected mechanisms used in deep learning such as smart initialization, annealed learning rate, layerwise pretraining, and noise injection (as done in dropout and SGD) arise naturally and automatically from this framework, without manually crafting them into the algorithms. Second, we present some preliminary results on comparing the proposed method against SGD. It turns out that the new algorithm can achieve similar level of generalization accuracy of SGD in much fewer number of epochs.
Very recently, the diffusion process has been proposed for learning difficult probability distributions @cite_5 . In forward time diffusion, the method converts any complex distribution into a simple distribution, e.g., Gaussian. It then learns the reverse-time of this diffusion process to define a generative model distribution. By sampling from such trained model, the authors have achieved inpainting of missing regions in natural images.
@cite_5
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Estimation of parameters of a diffusion based on discrete time observations poses a difficult problem due to the lack of a closed form expression for the likelihood. From a Bayesian computational perspective it can be casted as a missing data problem where the diffusion bridges in between discrete-time observations are missing. The computational problem can then be dealt with using a Markov-chain Monte-Carlo method known as data-augmentation. If unknown parameters appear in the diffusion coefficient, direct implementation of data-augmentation results in a Markov chain that is reducible. Furthermore, data-augmentation requires efficient sampling of diffusion bridges, which can be difficult, especially in the multidimensional case. We present a general framework to deal with with these problems that does not rely on discretisation. The construction generalises previous approaches and sheds light on the assumptions necessary to make these approaches work. We define a random-walk type Metropolis-Hastings sampler for updating diffusion bridges. Our methods are illustrated using guided proposals for sampling diffusion bridges. These are Markov processes obtained by adding a guiding term to the drift of the diffusion. We give general guidelines on the construction of these proposals and introduce a time change and scaling of the guided proposal that reduces discretisation error. Numerical examples demonstrate the performance of our methods. A central problem in machine learning involves modeling complex data-sets using highly flexible families of probability distributions in which learning, sampling, inference, and evaluation are still analytically or computationally tractable. Here, we develop an approach that simultaneously achieves both flexibility and tractability. The essential idea, inspired by non-equilibrium statistical physics, is to systematically and slowly destroy structure in a data distribution through an iterative forward diffusion process. We then learn a reverse diffusion process that restores structure in data, yielding a highly flexible and tractable generative model of the data. This approach allows us to rapidly learn, sample from, and evaluate probabilities in deep generative models with thousands of layers or time steps, as well as to compute conditional and posterior probabilities under the learned model. We additionally release an open source reference implementation of the algorithm. We show that a generative random field model, which we call generative ConvNet, can be derived from the commonly used discriminative ConvNet, by assuming a ConvNet for multicategory classification and assuming one of the categories is a base category generated by a reference distribution. If we further assume that the non-linearity in the ConvNet is Rectified Linear Unit (ReLU) and the reference distribution is Gaussian white noise, then we obtain a generative ConvNet model that is unique among energy-based models: The model is piecewise Gaussian, and the means of the Gaussian pieces are defined by an auto-encoder, where the filters in the bottom-up encoding become the basis functions in the top-down decoding, and the binary activation variables detected by the filters in the bottom-up convolution process become the coefficients of the basis functions in the top-down deconvolution process. The Langevin dynamics for sampling the generative ConvNet is driven by the reconstruction error of this autoencoder. The contrastive divergence learning of the generative ConvNet reconstructs the training images by the auto-encoder. The maximum likelihood learning algorithm can synthesize realistic natural image patterns. We show that a generative random field model, which we call generative ConvNet, can be derived from the commonly used discriminative ConvNet, by assuming a ConvNet for multi-category classification and assuming one of the categories is a base category generated by a reference distribution. If we further assume that the non-linearity in the ConvNet is Rectified Linear Unit (ReLU) and the reference distribution is Gaussian white noise, then we obtain a generative ConvNet model that is unique among energy-based models: The model is piecewise Gaussian, and the means of the Gaussian pieces are defined by an auto-encoder, where the filters in the bottom-up encoding become the basis functions in the top-down decoding, and the binary activation variables detected by the filters in the bottom-up convolution process become the coefficients of the basis functions in the top-down deconvolution process. The Langevin dynamics for sampling the generative ConvNet is driven by the reconstruction error of this auto-encoder. The contrastive divergence learning of the generative ConvNet reconstructs the training images by the auto-encoder. The maximum likelihood learning algorithm can synthesize realistic natural image patterns.