Auto-Encoders, Distributed Training and Information Representation in Deep Neural Networks
The goal of this thesis is to present a body of work that serves as my modest contribution to humanity's quest to understand intelligence and to implement intelligent systems. This is a thesis by articles, containing five articles, not all of equal impact, but all representing a very meaningful personal endeavor.  The articles are presented in chronological order, and they cluster around two general topics : representation learning and optimization. Articles from chapters 3, 5, and 9 are in the former category, whereas articles from chapters 7 and 11 are in the latter.  In the first article, we start with the idea of manifold learning through training a denoising auto-encoder to locally reconstruct data after perturbations. We establish a connection between contractive auto-encoders and denoising auto-encoders. More importantly, we prove mathematically a very interesting property from the optimal solution to denoising auto-encoders with additive gaussian noise. Namely, the fact that they learn exactly the score of the probability density function of the training distribution. We present a collection of ways in which this allows us to turn an auto-encoder into a generative model. We provide experiments all related to the goal of local manifold learning.  In the second article, we continue with that idea of building a generative model by learning conditional distributions. We do that in a more general setting and we focus more on the properties of the Markov chain obtained by Gibbs sampling. With a small modification in the construction of the Markov chain, we obtain the more general "Generative Stochastic Networks", which we can then stack together into a structure that can represent more accurately the different levels of abstraction of the data modeled. We present experiments involving the generation of MNIST digits and image inpainting.  In the third article, we present a novel idea for distributed optimization. Our proposal uses a collection of worker nodes to compute the importance weights to be used by one master node to perform Importance Sampling. This paradigm has a lot in common with the idea of curriculum learning, whereby the order of training examples is taken to have a significant impact on the training performance. We present results to compare the potential reduction in variance for gradient estimates with the practical reduction in variance observed.  In the fourth article, we go back to the concept of representation learning by asking whether there would be any measurable quantity in a neural network layer that would correspond intuitively to its "information contents". This is particularly interesting because there is a kind of paradox in deterministic neural networks : deeper layers encode better representations of the input signal, but they carry less (or equal) information than the raw inputs (in terms of entropy). By training a linear classifier on every layer in a neural network (with frozen parameters), we are able to measure linearly separability of the representations at every layer. We call these "linear classifier probes", and we show how they can be used to better understand the dynamics of training a neural network. We present experiments with large models (Inception v3 and ResNet-50) and uncover a surprizing property : linear separability increases in a strictly monotonic relationship with the layer depth.  In the fifth article, we revisit optimization again, but now we study the negative curvature of the loss function. We look at the most dominant eigenvalues and eigenvectors of the Hessian matrix, and we explore the gains to be made by modifying the model parameters along that direction with an optimal step size. We are mainly interested in the potential gains for directions of negative curvature, because those are ignored by the very popular convex optimization methods used by the deep learning community. Due to the large computational costs of anything dealing with the Hessian matrix, we run a small model on MNIST. We find that large gains can be made in directions of negative curvature, and that the optimal step sizes involved are larger than the current literature would recommend.
Deep learning Neural networks Representation learning Denoising auto-encoders Non-convex optimization