When optimizing over-parameterized models, such as deep neural networks, a large set of parameters leads to a zero training error. However they lead to different values for the test error and thus have distinct generalization properties. More specifically, Neyshabur [2017, Part II] argues that the choice of the optimization algorithm (and its respective hyperparameters) provides an implicit regularization with respect to its geometry: it biases the training, finding a particular minimizer of the objective. In this work, we use the same setting as Saxe et al. [2018]: a regression problem with least-square loss on a multidimensional output. Our prediction is made either by a linear model or by a two-layer linear neural network [Saxe et al., 2018]. Our goal is to extend their work on the continuous gradient dynamics in order to understand the behavior of the discrete dynamics induced by these two models. We show that with a vanishing initialization and a small enough step-size, the gradient dynamics of the two-layer linear neural network sequentially learns components that can be ranked according to a hierarchical structure whereas the gradient dynamics of the linear model learns the same components at the same time, missing this notion of hierarchy between components. The path followed by the two-layer formulation actually corresponds to successively solving the initial regression problem with a growing low rank constraint which is also know as reduced-rank regression [Izenman, 1975]. Note that this notion of path followed by the dynamics of a whole network is different from the notion of path introduced by Neyshabur et al. [2015a] which

Testing of deep learning models is challenging due to the excessive number and complexity of computations involved. As a result, test data selection is performed manually and in an ad hoc way. This raises the question of how we can automatically select candidate test data to test deep learning models. Recent research has focused on adapting test selection metrics from code-based software testing (such as coverage) to deep learning. However, deep learning models have different attributes from code such as spread of computations across the entire network reflecting training data properties, balance of neuron weights and redundancy (use of many more neurons than needed). Such differences make code-based metrics inappropriate to select data that can challenge the models (can trigger misclassification). We thus propose a set of test selection metrics based on the notion of model uncertainty (model confidence on specific inputs). Intuitively, the more uncertain we are about a candidate sample, the more likely it is that this sample triggers a misclassification. Similarly, the samples for which we are the most uncertain, are the most informative and should be used to improve the model by retraining. We evaluate these metrics on two widely-used image classification problems involving real and artificial (adversarial) data. We show that uncertainty-based metrics have a strong ability to select data that are misclassified and lead to major improvement in classification accuracy during retraining: up to 80% more gain than random selection and other state-of-the-art metrics on one dataset and up to 29% on the other.

Graph neural networks, which generalize deep neural network models to graph structured data, have attracted increasing attention in recent years. They usually learn node representations by transforming, propagating and aggregating node features and have been proven to improve the performance of many graph related tasks such as node classification and link prediction. To apply graph neural networks for the graph classification task, approaches to generate the \textit{graph representation} from node representations are demanded. A common way is to globally combine the node representations. However, rich structural information is overlooked. Thus a hierarchical pooling procedure is desired to preserve the graph structure during the graph representation learning. There are some recent works on hierarchically learning graph representation analogous to the pooling step in conventional convolutional neural (CNN) networks. However, the local structural information is still largely neglected during the pooling process. In this paper, we introduce a pooling operator $\pooling$ based on graph Fourier transform, which can utilize the node features and local structures during the pooling process. We then design pooling layers based on the pooling operator, which are further combined with traditional GCN convolutional layers to form a graph neural network framework $\m$ for graph classification. Theoretical analysis is provided to understand $\pooling$ from both local and global perspectives. Experimental results of the graph classification task on $6$ commonly used benchmarks demonstrate the effectiveness of the proposed framework.

The energy output a photo voltaic(PV) panel is a function of solar irradiation and weather parameters like temperature and wind speed etc. A general measure for solar irradiation called Global Horizontal Irradiance (GHI), customarily reported in Watt/meter$^2$, is a generic indicator for this intermittent energy resource. An accurate prediction of GHI is necessary for reliable grid integration of the renewable as well as for power market trading. While some machine learning techniques are well introduced along with the traditional time-series forecasting techniques, deep-learning techniques remains less explored for the task at hand. In this paper we give deep learning models suitable for sequence to sequence prediction of GHI. The deep learning models are reported for short-term forecasting $\{1-24\}$ hour along with the state-of-the art techniques like Gradient Boosted Regression Trees(GBRT) and Feed Forward Neural Networks(FFNN). We have checked that spatio-temporal features like wind direction, wind speed and GHI of neighboring location improves the prediction accuracy of the deep learning models significantly. Among the various sequence-to-sequence encoder-decoder models LSTM performed superior, handling short-comings of the state-of-the-art techniques.

Permutation of any two hidden units yields invariant properties in typical deep generative neural networks. This permutation symmetry plays an important role in understanding the computation performance of a broad class of neural networks with two or more hidden units. However, a theoretical study of the permutation symmetry is still lacking. Here, we propose a minimal model with only two hidden units in a restricted Boltzmann machine, which aims to address how the permutation symmetry affects the critical learning data size at which the concept-formation (or spontaneous symmetry breaking in physics language) starts, and moreover semi-rigorously prove a conjecture that the critical data size is independent of the number of hidden units once this number is finite. Remarkably, we find that the embedded correlation between two receptive fields of hidden units reduces the critical data size. In particular, the weakly-correlated receptive fields have the benefit of significantly reducing the minimal data size that triggers the transition, given less noisy data. Inspired by the theory, we also propose an efficient fully-distributed algorithm to infer the receptive fields of hidden units. Overall, our results demonstrate that the permutation symmetry is an interesting property that affects the critical data size for computation performances of related learning algorithms. All these effects can be analytically probed based on the minimal model, providing theoretical insights towards understanding unsupervised learning in a more general context.

Arguably, unsupervised learning plays a crucial role in the majority of algorithms for processing brain imaging. A recently introduced unsupervised approach Deep InfoMax (DIM) is a promising tool for exploring brain structure in a flexible non-linear way. In this paper, we investigate the use of variants of DIM in a setting of progression to Alzheimer's disease in comparison with supervised AlexNet and ResNet inspired convolutional neural networks. As a benchmark, we use a classification task between four groups: patients with stable, and progressive mild cognitive impairment (MCI), with Alzheimer's disease, and healthy controls. Our dataset is comprised of 828 subjects from the Alzheimer's Disease Neuroimaging Initiative (ADNI) database. Our experiments highlight encouraging evidence of the high potential utility of DIM in future neuroimaging studies.

Unsupervised domain adaptation of speech signal aims at adapting a well-trained source-domain acoustic model to the unlabeled data from target domain. This can be achieved by adversarial training of deep neural network (DNN) acoustic models to learn an intermediate deep representation that is both senone-discriminative and domain-invariant. Specifically, the DNN is trained to jointly optimize the primary task of senone classification and the secondary task of domain classification with adversarial objective functions. In this work, instead of only focusing on learning a domain-invariant feature (i.e. the shared component between domains), we also characterize the difference between the source and target domain distributions by explicitly modeling the private component of each domain through a private component extractor DNN. The private component is trained to be orthogonal with the shared component and thus implicitly increases the degree of domain-invariance of the shared component. A reconstructor DNN is used to reconstruct the original speech feature from the private and shared components as a regularization. This domain separation framework is applied to the unsupervised environment adaptation task and achieved 11.08% relative WER reduction from the gradient reversal layer training, a representative adversarial training method, for automatic speech recognition on CHiME-3 dataset.

Neural networks are increasingly used in complex (data-driven) simulations as surrogates or for accelerating the computation of classical surrogates. In many applications physical constraints, such as mass or energy conservation, must be satisfied to obtain reliable results. However, standard machine learning algorithms are generally not tailored to respect such constraints. We propose two different strategies to generate constraint-aware neural networks. We test their performance in the context of front-capturing schemes for strongly nonlinear wave motion in compressible fluid flow. Precisely, in this context so-called Riemann problems have to be solved as surrogates. Their solution describes the local dynamics of the captured wave front in numerical simulations. Three model problems are considered: a cubic flux model problem, an isothermal two-phase flow model, and the Euler equations. We demonstrate that a decrease in the constraint deviation correlates with low discretization errors for all model problems, in addition to the structural advantage of fulfilling the constraint.

The large thermal capacity of buildings enables heating, ventilating, and air-conditioning (HVAC) systems to be exploited as demand response (DR) resources. Optimal DR of HVAC units is challenging, particularly for multi-zone buildings, because this requires detailed physics-based models of zonal temperature variations for HVAC system operation and building thermal conditions. This paper proposes a new strategy for optimal DR of an HVAC system in a multi-zone building, based on supervised learning (SL). Artificial neural networks (ANNs) are trained with data obtained under normal building operating conditions. The ANNs are replicated using piecewise linear equations, which are explicitly integrated into an optimal scheduling problem for price-based DR. The optimization problem is solved for various electricity prices and building thermal conditions. The solutions are further used to train a deep neural network (DNN) to directly determine the optimal DR schedule, referred to here as supervised-learning-aided meta-prediction (SLAMP). Case studies are performed using three different methods: explicit ANN replication (EAR), SLAMP, and physics-based modeling. The case study results verify the effectiveness of the proposed SL-based strategy, in terms of both practical applicability and computational time, while also ensuring the thermal comfort of occupants and cost-effective operation of the HVAC system.

There is a stark disparity between the step size schedules used in practical large scale machine learning and those that are considered optimal by the theory of stochastic approximation. In theory, most results utilize polynomially decaying learning rate schedules, while, in practice, the "Step Decay" schedule is among the most popular schedules, where the learning rate is cut every constant number of epochs (i.e. this is a geometrically decaying schedule). This work examines the step-decay schedule for the stochastic optimization problem of streaming least squares regression (both in the non-strongly convex and strongly convex case), where we show that a sharp theoretical characterization of an optimal learning rate schedule is far more nuanced than suggested by previous work. We focus specifically on the rate that is achievable when using the final iterate of stochastic gradient descent, as is commonly done in practice. Our main result provably shows that a properly tuned geometrically decaying learning rate schedule provides an exponential improvement (in terms of the condition number) over any polynomially decaying learning rate schedule. We also provide experimental support for wider applicability of these results, including for training modern deep neural networks.