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Past research has indicated that the covariance of the Stochastic Gradient Descent (SGD) error done via minibatching plays a critical role in determining its regularization and escape from low potential points. Motivated by some new research in this area, we prove universality results by showing that noise classes that have the same mean and covariance structure of SGD via minibatching have similar properties. We mainly consider the Multiplicative Stochastic Gradient Descent (M-SGD) algorithm as introduced in previous work, which has a much more general noise class than the SGD algorithm done via minibatching. We establish non asymptotic bounds for the M-SGD algorithm in the Wasserstein distance. We also show that the M-SGD error is approximately a scaled Gaussian distribution with mean $0$ at any fixed point of the M-SGD algorithm.

Neural networks can be trained to solve regression problems by using gradient-based methods to minimize the square loss. However, practitioners often prefer to reformulate regression as a classification problem, observing that training on the cross entropy loss results in better performance. By focusing on two-layer ReLU networks, which can be fully characterized by measures over their feature space, we explore how the implicit bias induced by gradient-based optimization could partly explain the above phenomenon. We provide theoretical evidence that the regression formulation yields a measure whose support can differ greatly from that for classification, in the case of one-dimensional data. Our proposed optimal supports correspond directly to the features learned by the input layer of the network. The different nature of these supports sheds light on possible optimization difficulties the square loss could encounter during training, and we present empirical results illustrating this phenomenon.

Oversmoothing is a central challenge of building more powerful Graph Neural Networks (GNNs). While previous works have only demonstrated that oversmoothing is inevitable when the number of graph convolutions tends to infinity, in this paper, we precisely characterize the mechanism behind the phenomenon via a non-asymptotic analysis. Specifically, we distinguish between two different effects when applying graph convolutions -- an undesirable mixing effect that homogenizes node representations in different classes, and a desirable denoising effect that homogenizes node representations in the same class. By quantifying these two effects on random graphs sampled from the Contextual Stochastic Block Model (CSBM), we show that oversmoothing happens once the mixing effect starts to dominate the denoising effect, and the number of layers required for this transition is $O(\log N/\log (\log N))$ for sufficiently dense graphs with $N$ nodes. We also extend our analysis to study the effects of Personalized PageRank (PPR), or equivalently, the effects of initial residual connections on oversmoothing. Our results suggest that while PPR mitigates oversmoothing at deeper layers, PPR-based architectures still achieve their best performance at a shallow depth and are outperformed by the graph convolution approach on certain graphs. Finally, we support our theoretical results with numerical experiments, which further suggest that the oversmoothing phenomenon observed in practice can be magnified by the difficulty of optimizing deep GNN models.

The August 2022 special election for U.S. House Representative in Alaska featured three main candidates and was conducted by by single-winner ranked choice voting method known as ``instant runoff voting." The results of this election displayed a well-known but relatively rare phenomenon known as the ``center squeeze:" The most centrist candidate, Mark Begich, was eliminated in the first round despite winning an overwhelming majority of second-place votes. In fact, Begich was the {\em Condorcet winner} of this election: Based on the cast vote record, he would have defeated both of the other two candidates in head-to-head contests, but he was eliminated in the first round of ballot counting due to receiving the fewest first-place votes. The purpose of this paper is to use the data in the cast vote record to explore the range of likely outcomes if this election had been conducted under two alternative voting methods: Approval Voting and STAR (``Score Then Automatic Runoff") Voting. We find that under the best assumptions available about voter behavior, the most likely outcomes are that Peltola would still have won the election under Approval Voting, while Begich would have won under STAR Voting.

Sparse graph recovery methods works well where the data follows their assumptions but often they are not designed for doing downstream probabilistic queries. This limits their adoption to only identifying connections among the input variables. On the other hand, the Probabilistic Graphical Models (PGMs) assumes an underlying base graph between variables and learns a distribution over them. PGM design choices are carefully made such that the inference & sampling algorithms are efficient. This brings in certain restrictions and often simplifying assumptions. In this work, we propose Neural Graph Revealers (NGRs), that are an attempt to efficiently merge the sparse graph recovery methods with PGMs into a single flow. The problem setting consists of an input data X with D features and M samples and the task is to recover a sparse graph showing connection between the features. NGRs view the neural networks as a `white box' or more specifically as a multitask learning framework. We introduce `Graph-constrained path norm' that NGRs leverage to learn a graphical model that captures complex non-linear functional dependencies between the features in the form of an undirected sparse graph. Furthermore, NGRs can handle multimodal inputs like images, text, categorical data, embeddings etc. which is not straightforward to incorporate in the existing methods. We show experimental results of doing sparse graph recovery and probabilistic inference on data from Gaussian graphical models and a multimodal infant mortality dataset by CDC.

To deploy and operate deep neural models in production, the quality of their predictions, which might be contaminated benignly or manipulated maliciously by input distributional deviations, must be monitored and assessed. Specifically, we study the case of monitoring the healthy operation of a deep neural network (DNN) receiving a stream of data, with the aim of detecting input distributional deviations over which the quality of the network's predictions is potentially damaged. Using selective prediction principles, we propose a distribution deviation detection method for DNNs. The proposed method is derived from a tight coverage generalization bound computed over a sample of instances drawn from the true underlying distribution. Based on this bound, our detector continuously monitors the operation of the network over a test window and fires off an alarm whenever a deviation is detected. This novel detection method consistently and significantly outperforms the state of the art with respect to the CIFAR-10 and ImageNet datasets, thus establishing a new performance bar for this task, while being substantially more efficient in time and space complexities.

The remarkable successes of neural networks in a huge variety of inverse problems have fueled their adoption in disciplines ranging from medical imaging to seismic analysis over the past decade. However, the high dimensionality of such inverse problems has simultaneously left current theory, which predicts that networks should scale exponentially in the dimension of the problem, unable to explain why the seemingly small networks used in these settings work as well as they do in practice. To reduce this gap between theory and practice, we provide a general method for bounding the complexity required for a neural network to approximate a H\"older (or uniformly) continuous function defined on a high-dimensional set with a low-complexity structure. The approach is based on the observation that the existence of a Johnson-Lindenstrauss embedding $A\in\mathbb{R}^{d\times D}$ of a given high-dimensional set $S\subset\mathbb{R}^D$ into a low dimensional cube $[-M,M]^d$ implies that for any H\"older (or uniformly) continuous function $f:S\to\mathbb{R}^p$, there exists a H\"older (or uniformly) continuous function $g:[-M,M]^d\to\mathbb{R}^p$ such that $g(Ax)=f(x)$ for all $x\in S$. Hence, if one has a neural network which approximates $g:[-M,M]^d\to\mathbb{R}^p$, then a layer can be added that implements the JL embedding $A$ to obtain a neural network that approximates $f:S\to\mathbb{R}^p$. By pairing JL embedding results along with results on approximation of H\"older (or uniformly) continuous functions by neural networks, one then obtains results which bound the complexity required for a neural network to approximate H\"older (or uniformly) continuous functions on high dimensional sets. The end result is a general theoretical framework which can then be used to better explain the observed empirical successes of smaller networks in a wider variety of inverse problems than current theory allows.

The time and effort involved in hand-designing deep neural networks is immense. This has prompted the development of Neural Architecture Search (NAS) techniques to automate this design. However, NAS algorithms tend to be slow and expensive; they need to train vast numbers of candidate networks to inform the search process. This could be alleviated if we could partially predict a network's trained accuracy from its initial state. In this work, we examine the overlap of activations between datapoints in untrained networks and motivate how this can give a measure which is usefully indicative of a network's trained performance. We incorporate this measure into a simple algorithm that allows us to search for powerful networks without any training in a matter of seconds on a single GPU, and verify its effectiveness on NAS-Bench-101, NAS-Bench-201, NATS-Bench, and Network Design Spaces. Our approach can be readily combined with more expensive search methods; we examine a simple adaptation of regularised evolutionary search. Code for reproducing our experiments is available at //github.com/BayesWatch/nas-without-training.

Current deep learning research is dominated by benchmark evaluation. A method is regarded as favorable if it empirically performs well on the dedicated test set. This mentality is seamlessly reflected in the resurfacing area of continual learning, where consecutively arriving sets of benchmark data are investigated. The core challenge is framed as protecting previously acquired representations from being catastrophically forgotten due to the iterative parameter updates. However, comparison of individual methods is nevertheless treated in isolation from real world application and typically judged by monitoring accumulated test set performance. The closed world assumption remains predominant. It is assumed that during deployment a model is guaranteed to encounter data that stems from the same distribution as used for training. This poses a massive challenge as neural networks are well known to provide overconfident false predictions on unknown instances and break down in the face of corrupted data. In this work we argue that notable lessons from open set recognition, the identification of statistically deviating data outside of the observed dataset, and the adjacent field of active learning, where data is incrementally queried such that the expected performance gain is maximized, are frequently overlooked in the deep learning era. Based on these forgotten lessons, we propose a consolidated view to bridge continual learning, active learning and open set recognition in deep neural networks. Our results show that this not only benefits each individual paradigm, but highlights the natural synergies in a common framework. We empirically demonstrate improvements when alleviating catastrophic forgetting, querying data in active learning, selecting task orders, while exhibiting robust open world application where previously proposed methods fail.

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.