We consider the problem of learning functions in the $\mathcal{F}_{p,\pi}$ and Barron spaces, which are natural function spaces that arise in the high-dimensional analysis of random feature models (RFMs) and two-layer neural networks. Through a duality analysis, we reveal that the approximation and estimation of these spaces can be considered equivalent in a certain sense. This enables us to focus on the easier problem of approximation and estimation when studying the generalization of both models. The dual equivalence is established by defining an information-based complexity that can effectively control estimation errors. Additionally, we demonstrate the flexibility of our duality framework through comprehensive analyses of two concrete applications. The first application is to study learning functions in $\mathcal{F}_{p,\pi}$ with RFMs. We prove that the learning does not suffer from the curse of dimensionality as long as $p>1$, implying RFMs can work beyond the kernel regime. Our analysis extends existing results [CMM21] to the noisy case and removes the requirement of overparameterization. The second application is to investigate the learnability of reproducing kernel Hilbert space (RKHS) under the $L^\infty$ metric. We derive both lower and upper bounds of the minimax estimation error by using the spectrum of the associated kernel. We then apply these bounds to dot-product kernels and analyze how they scale with the input dimension. Our results suggest that learning with ReLU (random) features is generally intractable in terms of reaching high uniform accuracy.
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R is a language and environment for statistical computing and graphics, which provides a wide variety of statistical tools (modeling, statistical testing, time series analysis, classification problems, machine learning, ...), together with amazing graphical techniques and the great advantage that it is highly extensible. Nowadays, there is no doubt that it is the software par excellence in statistical courses for any level, for theoretical and applied subjects alike. Besides, it has become an almost essential tool for every research work that involves any kind of analysis or data visualization. Furthermore, it is one of the most employed programming languages for general purposes. The goal of this work is helping to share ideas and resources to improve teaching and/or research using the statistical software R. We will cover its benefits, show how to get started and where to locate specific resources, and will make interesting recommendations for using R, according to our experience. For the classroom we will develop a curricular and assessment infrastructure to support both dissemination and evaluation, while for research we will offer a broader approach to quantitative studies that provides an excellent support for work in science and technology.
Stochastic optimization has found wide applications in minimizing objective functions in machine learning, which motivates a lot of theoretical studies to understand its practical success. Most of existing studies focus on the convergence of optimization errors, while the generalization analysis of stochastic optimization is much lagging behind. This is especially the case for nonconvex and nonsmooth problems often encountered in practice. In this paper, we initialize a systematic stability and generalization analysis of stochastic optimization on nonconvex and nonsmooth problems. We introduce novel algorithmic stability measures and establish their quantitative connection on the gap between population gradients and empirical gradients, which is then further extended to study the gap between the Moreau envelope of the empirical risk and that of the population risk. To our knowledge, these quantitative connection between stability and generalization in terms of either gradients or Moreau envelopes have not been studied in the literature. We introduce a class of sampling-determined algorithms, for which we develop bounds for three stability measures. Finally, we apply these discussions to derive error bounds for stochastic gradient descent and its adaptive variant, where we show how to achieve an implicit regularization by tuning the step sizes and the number of iterations.
Staging of liver fibrosis is important in the diagnosis and treatment planning of patients suffering from liver diseases. Current deep learning-based methods using abdominal magnetic resonance imaging (MRI) usually take a sub-region of the liver as an input, which nevertheless could miss critical information. To explore richer representations, we formulate this task as a multi-view learning problem and employ multiple sub-regions of the liver. Previously, features or predictions are usually combined in an implicit manner, and uncertainty-aware methods have been proposed. However, these methods could be challenged to capture cross-view representations, which can be important in the accurate prediction of staging. Therefore, we propose a reliable multi-view learning method with interpretable combination rules, which can model global representations to improve the accuracy of predictions. Specifically, the proposed method estimates uncertainties based on subjective logic to improve reliability, and an explicit combination rule is applied based on Dempster-Shafer's evidence theory with good power of interpretability. Moreover, a data-efficient transformer is introduced to capture representations in the global view. Results evaluated on enhanced MRI data show that our method delivers superior performance over existing multi-view learning methods.
Understanding the effect of a feature vector $x \in \mathbb{R}^d$ on the response value (label) $y \in \mathbb{R}$ is the cornerstone of many statistical learning problems. Ideally, it is desired to understand how a set of collected features combine together and influence the response value, but this problem is notoriously difficult, due to the high-dimensionality of data and limited number of labeled data points, among many others. In this work, we take a new perspective on this problem, and we study the question of assessing the difference of influence that the two given features have on the response value. We first propose a notion of closeness for the influence of features, and show that our definition recovers the familiar notion of the magnitude of coefficients in the parametric model. We then propose a novel method to test for the closeness of influence in general model-free supervised learning problems. Our proposed test can be used with finite number of samples with control on type I error rate, no matter the ground truth conditional law $\mathcal{L}(Y |X)$. We analyze the power of our test for two general learning problems i) linear regression, and ii) binary classification under mixture of Gaussian models, and show that under the proper choice of score function, an internal component of our test, with sufficient number of samples will achieve full statistical power. We evaluate our findings through extensive numerical simulations, specifically we adopt the datamodel framework (Ilyas, et al., 2022) for CIFAR-10 dataset to identify pairs of training samples with different influence on the trained model via optional black box training mechanisms.
To ensure resilient neural network processing on even unreliable hardware, comprehensive reliability analysis against various hardware faults is generally required before the deep neural network models are deployed, and efficient error injection tools are highly demanded. However, most existing fault injection tools remain rather limited to basic fault injection to neurons and fail to provide fine-grained vulnerability analysis capability. In addition, many of the fault injection tools still need to change the neural network models and make the fault injection closely coupled with normal neural network processing, which further complicates the use of the fault injection tools and slows down the fault simulation. In this work, we propose MRFI, a highly configurable multi-resolution fault injection tool for deep neural networks. It enables users to modify an independent fault configuration file rather than neural network models for the fault injection and vulnerability analysis. Particularly, it integrates extensive fault analysis functionalities from different perspectives and enables multi-resolution investigation of the vulnerability of neural networks. In addition, it does not modify the major neural network computing framework of PyTorch. Hence, it allows parallel processing on GPUs naturally and exhibits fast fault simulation according to our experiments.
Knowledge graph embedding (KGE) is a increasingly popular technique that aims to represent entities and relations of knowledge graphs into low-dimensional semantic spaces for a wide spectrum of applications such as link prediction, knowledge reasoning and knowledge completion. In this paper, we provide a systematic review of existing KGE techniques based on representation spaces. Particularly, we build a fine-grained classification to categorise the models based on three mathematical perspectives of the representation spaces: (1) Algebraic perspective, (2) Geometric perspective, and (3) Analytical perspective. We introduce the rigorous definitions of fundamental mathematical spaces before diving into KGE models and their mathematical properties. We further discuss different KGE methods over the three categories, as well as summarise how spatial advantages work over different embedding needs. By collating the experimental results from downstream tasks, we also explore the advantages of mathematical space in different scenarios and the reasons behind them. We further state some promising research directions from a representation space perspective, with which we hope to inspire researchers to design their KGE models as well as their related applications with more consideration of their mathematical space properties.
Denoising diffusion models represent a recent emerging topic in computer vision, demonstrating remarkable results in the area of generative modeling. A diffusion model is a deep generative model that is based on two stages, a forward diffusion stage and a reverse diffusion stage. In the forward diffusion stage, the input data is gradually perturbed over several steps by adding Gaussian noise. In the reverse stage, a model is tasked at recovering the original input data by learning to gradually reverse the diffusion process, step by step. Diffusion models are widely appreciated for the quality and diversity of the generated samples, despite their known computational burdens, i.e. low speeds due to the high number of steps involved during sampling. In this survey, we provide a comprehensive review of articles on denoising diffusion models applied in vision, comprising both theoretical and practical contributions in the field. First, we identify and present three generic diffusion modeling frameworks, which are based on denoising diffusion probabilistic models, noise conditioned score networks, and stochastic differential equations. We further discuss the relations between diffusion models and other deep generative models, including variational auto-encoders, generative adversarial networks, energy-based models, autoregressive models and normalizing flows. Then, we introduce a multi-perspective categorization of diffusion models applied in computer vision. Finally, we illustrate the current limitations of diffusion models and envision some interesting directions for future research.
Graph Neural Networks (GNNs) have received considerable attention on graph-structured data learning for a wide variety of tasks. The well-designed propagation mechanism which has been demonstrated effective is the most fundamental part of GNNs. Although most of GNNs basically follow a message passing manner, litter effort has been made to discover and analyze their essential relations. In this paper, we establish a surprising connection between different propagation mechanisms with a unified optimization problem, showing that despite the proliferation of various GNNs, in fact, their proposed propagation mechanisms are the optimal solution optimizing a feature fitting function over a wide class of graph kernels with a graph regularization term. Our proposed unified optimization framework, summarizing the commonalities between several of the most representative GNNs, not only provides a macroscopic view on surveying the relations between different GNNs, but also further opens up new opportunities for flexibly designing new GNNs. With the proposed framework, we discover that existing works usually utilize naive graph convolutional kernels for feature fitting function, and we further develop two novel objective functions considering adjustable graph kernels showing low-pass or high-pass filtering capabilities respectively. Moreover, we provide the convergence proofs and expressive power comparisons for the proposed models. Extensive experiments on benchmark datasets clearly show that the proposed GNNs not only outperform the state-of-the-art methods but also have good ability to alleviate over-smoothing, and further verify the feasibility for designing GNNs with our unified optimization framework.
The concept of smart grid has been introduced as a new vision of the conventional power grid to figure out an efficient way of integrating green and renewable energy technologies. In this way, Internet-connected smart grid, also called energy Internet, is also emerging as an innovative approach to ensure the energy from anywhere at any time. The ultimate goal of these developments is to build a sustainable society. However, integrating and coordinating a large number of growing connections can be a challenging issue for the traditional centralized grid system. Consequently, the smart grid is undergoing a transformation to the decentralized topology from its centralized form. On the other hand, blockchain has some excellent features which make it a promising application for smart grid paradigm. In this paper, we have an aim to provide a comprehensive survey on application of blockchain in smart grid. As such, we identify the significant security challenges of smart grid scenarios that can be addressed by blockchain. Then, we present a number of blockchain-based recent research works presented in different literatures addressing security issues in the area of smart grid. We also summarize several related practical projects, trials, and products that have been emerged recently. Finally, we discuss essential research challenges and future directions of applying blockchain to smart grid security issues.
In structure learning, the output is generally a structure that is used as supervision information to achieve good performance. Considering the interpretation of deep learning models has raised extended attention these years, it will be beneficial if we can learn an interpretable structure from deep learning models. In this paper, we focus on Recurrent Neural Networks (RNNs) whose inner mechanism is still not clearly understood. We find that Finite State Automaton (FSA) that processes sequential data has more interpretable inner mechanism and can be learned from RNNs as the interpretable structure. We propose two methods to learn FSA from RNN based on two different clustering methods. We first give the graphical illustration of FSA for human beings to follow, which shows the interpretability. From the FSA's point of view, we then analyze how the performance of RNNs are affected by the number of gates, as well as the semantic meaning behind the transition of numerical hidden states. Our results suggest that RNNs with simple gated structure such as Minimal Gated Unit (MGU) is more desirable and the transitions in FSA leading to specific classification result are associated with corresponding words which are understandable by human beings.
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