B. Mehlig

L. Storm, H. Linander, J. Bec et al.

We compute how small input perturbations affect the output of deep neural networks, exploring an analogy between deep networks and dynamical systems, where the growth or decay of local perturbations is characterised by finite-time Lyapunov exponents. We show that the maximal exponent forms geometrical structures in input space, akin to coherent structures in dynamical systems. Ridges of large positive exponents divide input space into different regions that the network associates with different classes. These ridges visualise the geometry that deep networks construct in input space, shedding light on the fundamental mechanisms underlying their learning capabilities.

Oleksandr Balabanov, Bernhard Mehlig, Hampus Linander

We introduce ensembles of stochastic neural networks to approximate the Bayesian posterior, combining stochastic methods such as dropout with deep ensembles. The stochastic ensembles are formulated as families of distributions and trained to approximate the Bayesian posterior with variational inference. We implement stochastic ensembles based on Monte Carlo dropout, DropConnect and a novel non-parametric version of dropout and evaluate them on a toy problem and CIFAR image classification. For both tasks, we test the quality of the posteriors directly against Hamiltonian Monte Carlo simulations. Our results show that stochastic ensembles provide more accurate posterior estimates than other popular baselines for Bayesian inference.

L. Storm, K. Gustavsson, B. Mehlig

Echo-state networks are simple models of discrete dynamical systems driven by a time series. By selecting network parameters such that the dynamics of the network is contractive, characterized by a negative maximal Lyapunov exponent, the network may synchronize with the driving signal. Exploiting this synchronization, the echo-state network may be trained to autonomously reproduce the input dynamics, enabling time-series prediction. However, while synchronization is a necessary condition for prediction, it is not sufficient. Here, we study what other conditions are necessary for successful time-series prediction. We identify two key parameters for prediction performance, and conduct a parameter sweep to find regions where prediction is successful. These regions differ significantly depending on whether full or partial phase space information about the input is provided to the network during training. We explain how these regions emerge.

H. Linander, O. Balabanov, H. Yang et al.

Bayesian inference can quantify uncertainty in the predictions of neural networks using posterior distributions for model parameters and network output. By looking at these posterior distributions, one can separate the origin of uncertainty into aleatoric and epistemic contributions. One goal of uncertainty quantification is to inform on prediction accuracy. Here we show that prediction accuracy depends on both epistemic and aleatoric uncertainty in an intricate fashion that cannot be understood in terms of marginalized uncertainty distributions alone. How the accuracy relates to epistemic and aleatoric uncertainties depends not only on the model architecture, but also on the properties of the dataset. We discuss the significance of these results for active learning and introduce a novel acquisition function that outperforms common uncertainty-based methods. To arrive at our results, we approximated the posteriors using deep ensembles, for fully-connected, convolutional and attention-based neural networks.

B. Mehlig

These are lecture notes for a course on machine learning with neural networks for scientists and engineers that I have given at Gothenburg University and Chalmers Technical University in Gothenburg, Sweden. The material is organised into three parts: Hopfield networks, supervised learning of labeled data, and learning algorithms for unlabeled data sets. Part I introduces stochastic recurrent networks: Hopfield networks and Boltzmann machines. The analysis of their learning rules sets the scene for the later parts. Part II describes supervised learning with multilayer perceptrons and convolutional neural networks. This part starts with a simple geometrical interpretation of the learning rule and leads to the recent successes of convolutional networks in object recognition, recurrent networks in language processing, and reservoir computers in time-series analysis. Part III explains what neural networks can learn about data that is not labeled. This part begins with a description of unsupervised learning techniques for clustering of data, non-linear projections, and embeddings. A section on autoencoders explains how to learn without labels using convolutional networks, and the last chapter is dedicated to reinforcement learning. The overall goal of the course is to explain the fundamental principles that allow neural networks to learn, emphasising ideas and concepts that are common to all three parts. The present version does not contain exercises (copyright owned by Cambridge University Press). The complete book is available at https://www.cambridge.org/gb/academic/subjects/physics/statistical-physics/machine-learning-neural-networks-introduction-scientists-and-engineers?format=HB.