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Multilayer networks are a network data structure in which elements in a population of interest have multiple modes of interaction or relation, represented by multiple networks called layers. We propose a novel class of models for cross-layer dependence in multilayer networks, aiming to learn how interactions in one or more layers may influence interactions in other layers of the multilayer network, by developing a class of network separable models which separate the network formation process from the layer formation process. In our framework, we are able to extend existing single layer network models to a multilayer network model with cross-layer dependence. We establish non-asymptotic bounds on the error of estimators and demonstrate rates of convergence for both maximum likelihood estimators and maximum pseudolikelihood estimators in scenarios of increasing parameter dimension. We additionally establish non-asymptotic error bounds on the multivariate normal approximation and elaborate a method for model selection which controls the false discovery rate. We conduct simulation studies which demonstrate that our framework and method work well in realistic settings which might be encountered in applications. Lastly, we illustrate the utility of our method through an application to the Lazega lawyers network.

Recent works show that the data distribution in a network's latent space is useful for estimating classification uncertainty and detecting Out-of-distribution (OOD) samples. To obtain a well-regularized latent space that is conducive for uncertainty estimation, existing methods bring in significant changes to model architectures and training procedures. In this paper, we present a lightweight, fast, and high-performance regularization method for Mahalanobis distance-based uncertainty prediction, and that requires minimal changes to the network's architecture. To derive Gaussian latent representation favourable for Mahalanobis Distance calculation, we introduce a self-supervised representation learning method that separates in-class representations into multiple Gaussians. Classes with non-Gaussian representations are automatically identified and dynamically clustered into multiple new classes that are approximately Gaussian. Evaluation on standard OOD benchmarks shows that our method achieves state-of-the-art results on OOD detection with minimal inference time, and is very competitive on predictive probability calibration. Finally, we show the applicability of our method to a real-life computer vision use case on microorganism classification.

In computational neuroscience, fixed points of recurrent neural networks are commonly used to model neural responses to static or slowly changing stimuli. These applications raise the question of how to train the weights in a recurrent neural network to minimize a loss function evaluated on fixed points. A natural approach is to use gradient descent on the Euclidean space of synaptic weights. We show that this approach can lead to poor learning performance due, in part, to singularities that arise in the loss surface. We use a reparameterization of the recurrent network model to derive two alternative learning rules that produces more robust learning dynamics. We show that these learning rules can be interpreted as steepest descent and gradient descent, respectively, under a non-Euclidean metric on the space of recurrent weights. Our results question the common, implicit assumption that learning in the brain should be expected to follow the negative Euclidean gradient of synaptic weights.

The ever-growing use of wind energy makes necessary the optimization of turbine operations through pitch angle controllers and their maintenance with early fault detection. It is crucial to have accurate and robust models imitating the behavior of wind turbines, especially to predict the generated power as a function of the wind speed. Existing empirical and physics-based models have limitations in capturing the complex relations between the input variables and the power, aggravated by wind variability. Data-driven methods offer new opportunities to enhance wind turbine modeling of large datasets by improving accuracy and efficiency. In this study, we used physics-informed neural networks to reproduce historical data coming from 4 turbines in a wind farm, while imposing certain physical constraints to the model. The developed models for regression of the power, torque, and power coefficient as output variables showed great accuracy for both real data and physical equations governing the system. Lastly, introducing an efficient evidential layer provided uncertainty estimations of the predictions, proved to be consistent with the absolute error, and made possible the definition of a confidence interval in the power curve.

In non-smooth stochastic optimization, we establish the non-convergence of the stochastic subgradient descent (SGD) to the critical points recently called active strict saddles by Davis and Drusvyatskiy. Such points lie on a manifold $M$ where the function $f$ has a direction of second-order negative curvature. Off this manifold, the norm of the Clarke subdifferential of $f$ is lower-bounded. We require two conditions on $f$. The first assumption is a Verdier stratification condition, which is a refinement of the popular Whitney stratification. It allows us to establish a reinforced version of the projection formula of Bolte \emph{et.al.} for Whitney stratifiable functions, and which is of independent interest. The second assumption, termed the angle condition, allows to control the distance of the iterates to $M$. When $f$ is weakly convex, our assumptions are generic. Consequently, generically in the class of definable weakly convex functions, the SGD converges to a local minimizer.

Attention models are typically learned by optimizing one of three standard loss functions that are variously called -- soft attention, hard attention, and latent variable marginal likelihood (LVML) attention. All three paradigms are motivated by the same goal of finding two models -- a `focus' model that `selects' the right \textit{segment} of the input and a `classification' model that processes the selected segment into the target label. However, they differ significantly in the way the selected segments are aggregated, resulting in distinct dynamics and final results. We observe a unique signature of models learned using these paradigms and explain this as a consequence of the evolution of the classification model under gradient descent when the focus model is fixed. We also analyze these paradigms in a simple setting and derive closed-form expressions for the parameter trajectory under gradient flow. With the soft attention loss, the focus model improves quickly at initialization and splutters later on. On the other hand, hard attention loss behaves in the opposite fashion. Based on our observations, we propose a simple hybrid approach that combines the advantages of the different loss functions and demonstrates it on a collection of semi-synthetic and real-world datasets

Representation learning plays a crucial role in automated feature selection, particularly in the context of high-dimensional data, where non-parametric methods often struggle. In this study, we focus on supervised learning scenarios where the pertinent information resides within a lower-dimensional linear subspace of the data, namely the multi-index model. If this subspace were known, it would greatly enhance prediction, computation, and interpretation. To address this challenge, we propose a novel method for linear feature learning with non-parametric prediction, which simultaneously estimates the prediction function and the linear subspace. Our approach employs empirical risk minimisation, augmented with a penalty on function derivatives, ensuring versatility. Leveraging the orthogonality and rotation invariance properties of Hermite polynomials, we introduce our estimator, named RegFeaL. By utilising alternative minimisation, we iteratively rotate the data to improve alignment with leading directions and accurately estimate the relevant dimension in practical settings. We establish that our method yields a consistent estimator of the prediction function with explicit rates. Additionally, we provide empirical results demonstrating the performance of RegFeaL in various experiments.

This book develops an effective theory approach to understanding deep neural networks of practical relevance. Beginning from a first-principles component-level picture of networks, we explain how to determine an accurate description of the output of trained networks by solving layer-to-layer iteration equations and nonlinear learning dynamics. A main result is that the predictions of networks are described by nearly-Gaussian distributions, with the depth-to-width aspect ratio of the network controlling the deviations from the infinite-width Gaussian description. We explain how these effectively-deep networks learn nontrivial representations from training and more broadly analyze the mechanism of representation learning for nonlinear models. From a nearly-kernel-methods perspective, we find that the dependence of such models' predictions on the underlying learning algorithm can be expressed in a simple and universal way. To obtain these results, we develop the notion of representation group flow (RG flow) to characterize the propagation of signals through the network. By tuning networks to criticality, we give a practical solution to the exploding and vanishing gradient problem. We further explain how RG flow leads to near-universal behavior and lets us categorize networks built from different activation functions into universality classes. Altogether, we show that the depth-to-width ratio governs the effective model complexity of the ensemble of trained networks. By using information-theoretic techniques, we estimate the optimal aspect ratio at which we expect the network to be practically most useful and show how residual connections can be used to push this scale to arbitrary depths. With these tools, we can learn in detail about the inductive bias of architectures, hyperparameters, and optimizers.

Multi-Task Learning (MTL) is a learning paradigm in machine learning and its aim is to leverage useful information contained in multiple related tasks to help improve the generalization performance of all the tasks. In this paper, we give a survey for MTL from the perspective of algorithmic modeling, applications and theoretical analyses. For algorithmic modeling, we give a definition of MTL and then classify different MTL algorithms into five categories, including feature learning approach, low-rank approach, task clustering approach, task relation learning approach and decomposition approach as well as discussing the characteristics of each approach. In order to improve the performance of learning tasks further, MTL can be combined with other learning paradigms including semi-supervised learning, active learning, unsupervised learning, reinforcement learning, multi-view learning and graphical models. When the number of tasks is large or the data dimensionality is high, we review online, parallel and distributed MTL models as well as dimensionality reduction and feature hashing to reveal their computational and storage advantages. Many real-world applications use MTL to boost their performance and we review representative works in this paper. Finally, we present theoretical analyses and discuss several future directions for MTL.

For deploying a deep learning model into production, it needs to be both accurate and compact to meet the latency and memory constraints. This usually results in a network that is deep (to ensure performance) and yet thin (to improve computational efficiency). In this paper, we propose an efficient method to train a deep thin network with a theoretic guarantee. Our method is motivated by model compression. It consists of three stages. In the first stage, we sufficiently widen the deep thin network and train it until convergence. In the second stage, we use this well-trained deep wide network to warm up (or initialize) the original deep thin network. This is achieved by letting the thin network imitate the immediate outputs of the wide network from layer to layer. In the last stage, we further fine tune this well initialized deep thin network. The theoretical guarantee is established by using mean field analysis, which shows the advantage of layerwise imitation over traditional training deep thin networks from scratch by backpropagation. We also conduct large-scale empirical experiments to validate our approach. By training with our method, ResNet50 can outperform ResNet101, and BERT_BASE can be comparable with BERT_LARGE, where both the latter models are trained via the standard training procedures as in the literature.