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Adversarial training can be used to learn models that are robust against perturbations. For linear models, it can be formulated as a convex optimization problem. Compared to methods proposed in the context of deep learning, leveraging the optimization structure allows significantly faster convergence rates. Still, the use of generic convex solvers can be inefficient for large-scale problems. Here, we propose tailored optimization algorithms for the adversarial training of linear models, which render large-scale regression and classification problems more tractable. For regression problems, we propose a family of solvers based on iterative ridge regression and, for classification, a family of solvers based on projected gradient descent. The methods are based on extended variable reformulations of the original problem. We illustrate their efficiency in numerical examples.

State-of-the-art machine learning models can be vulnerable to very small input perturbations that are adversarially constructed. Adversarial training is an effective approach to defend against it. Formulated as a min-max problem, it searches for the best solution when the training data were corrupted by the worst-case attacks. Linear models are among the simple models where vulnerabilities can be observed and are the focus of our study. In this case, adversarial training leads to a convex optimization problem which can be formulated as the minimization of a finite sum. We provide a comparative analysis between the solution of adversarial training in linear regression and other regularization methods. Our main findings are that: (A) Adversarial training yields the minimum-norm interpolating solution in the overparameterized regime (more parameters than data), as long as the maximum disturbance radius is smaller than a threshold. And, conversely, the minimum-norm interpolator is the solution to adversarial training with a given radius. (B) Adversarial training can be equivalent to parameter shrinking methods (ridge regression and Lasso). This happens in the underparametrized region, for an appropriate choice of adversarial radius and zero-mean symmetrically distributed covariates. (C) For $\ell_\infty$-adversarial training -- as in square-root Lasso -- the choice of adversarial radius for optimal bounds does not depend on the additive noise variance. We confirm our theoretical findings with numerical examples., Comment: Accepted (spotlight) NeurIPS 2023; A preliminary version of this work titled: "Surprises in adversarially-trained linear regression" was made available under a different identifier: arXiv:2205.12695

Background: Atrial fibrillation (AF) is one of the most common cardiac arrhythmias that affects millions of people each year worldwide and it is closely linked to increased risk of cardiovascular diseases such as stroke and heart failure. Machine learning methods have shown promising results in evaluating the risk of developing atrial fibrillation from the electrocardiogram. We aim to develop and evaluate one such algorithm on a large CODE dataset collected in Brazil. Results: The deep neural network model identified patients without indication of AF in the presented ECG but who will develop AF in the future with an AUC score of 0.845. From our survival model, we obtain that patients in the high-risk group (i.e. with the probability of a future AF case being greater than 0.7) are 50% more likely to develop AF within 40 weeks, while patients belonging to the minimal-risk group (i.e. with the probability of a future AF case being less than or equal to 0.1) have more than 85% chance of remaining AF free up until after seven years. Conclusion: We developed and validated a model for AF risk prediction. If applied in clinical practice, the model possesses the potential of providing valuable and useful information in decision-making and patient management processes., Comment: 16 pages with 7 figures

Deep learning is a topic of considerable current interest. The availability of massive data collections and powerful software resources has led to an impressive amount of results in many application areas that reveal essential but hidden properties of the observations. System identification learns mathematical descriptions of dynamic systems from input-output data and can thus benefit from the advances of deep neural networks to enrich the possible range of models to choose from. For this reason, we provide a survey of deep learning from a system identification perspective. We cover a wide spectrum of topics to enable researchers to understand the methods, providing rigorous practical and theoretical insights into the benefits and challenges of using them. The main aim of the identified model is to predict new data from previous observations. This can be achieved with different deep learning based modelling techniques and we discuss architectures commonly adopted in the literature, like feedforward, convolutional, and recurrent networks. Their parameters have to be estimated from past data trying to optimize the prediction performance. For this purpose, we discuss a specific set of first-order optimization tools that is emerged as efficient. The survey then draws connections to the well-studied area of kernel-based methods. They control the data fit by regularization terms that penalize models not in line with prior assumptions. We illustrate how to cast them in deep architectures to obtain deep kernel-based methods. The success of deep learning also resulted in surprising empirical observations, like the counter-intuitive behaviour of models with many parameters. We discuss the role of overparameterized models, including their connection to kernels, as well as implicit regularization mechanisms which affect generalization, specifically the interesting phenomena of benign overfitting ...

Objective: Imbalances of the electrolyte concentration levels in the body can lead to catastrophic consequences, but accurate and accessible measurements could improve patient outcomes. While blood tests provide accurate measurements, they are invasive and the laboratory analysis can be slow or inaccessible. In contrast, an electrocardiogram (ECG) is a widely adopted tool which is quick and simple to acquire. However, the problem of estimating continuous electrolyte concentrations directly from ECGs is not well-studied. We therefore investigate if regression methods can be used for accurate ECG-based prediction of electrolyte concentrations. Methods: We explore the use of deep neural networks (DNNs) for this task. We analyze the regression performance across four electrolytes, utilizing a novel dataset containing over 290000 ECGs. For improved understanding, we also study the full spectrum from continuous predictions to binary classification of extreme concentration levels. To enhance clinical usefulness, we finally extend to a probabilistic regression approach and evaluate different uncertainty estimates. Results: We find that the performance varies significantly between different electrolytes, which is clinically justified in the interplay of electrolytes and their manifestation in the ECG. We also compare the regression accuracy with that of traditional machine learning models, demonstrating superior performance of DNNs. Conclusion: Discretization can lead to good classification performance, but does not help solve the original problem of predicting continuous concentration levels. While probabilistic regression demonstrates potential practical usefulness, the uncertainty estimates are not particularly well-calibrated. Significance: Our study is a first step towards accurate and reliable ECG-based prediction of electrolyte concentration levels., Comment: Code and trained models are available at https://github.com/philippvb/ecg-electrolyte-regression

Objective: Machine learning techniques have been used extensively for 12-lead electrocardiogram (ECG) analysis. For physiological time series, deep learning (DL) superiority to feature engineering (FE) approaches based on domain knowledge is still an open question. Moreover, it remains unclear whether combining DL with FE may improve performance. Methods: We considered three tasks intending to address these research gaps: cardiac arrhythmia diagnosis (multiclass-multilabel classification), atrial fibrillation risk prediction (binary classification), and age estimation (regression). We used an overall dataset of 2.3M 12-lead ECG recordings to train the following models for each task: i) a random forest taking the FE as input was trained as a classical machine learning approach; ii) an end-to-end DL model; and iii) a merged model of FE+DL. Results: FE yielded comparable results to DL while necessitating significantly less data for the two classification tasks and it was outperformed by DL for the regression task. For all tasks, merging FE with DL did not improve performance over DL alone. Conclusion: We found that for traditional 12-lead ECG based diagnosis tasks DL did not yield a meaningful improvement over FE, while it improved significantly the nontraditional regression task. We also found that combining FE with DL did not improve over DL alone which suggests that the FE were redundant with the features learned by DL. Significance: Our findings provides important recommendations on what machine learning strategy and data regime to chose with respect to the task at hand for the development of new machine learning models based on the 12-lead ECG.

State-of-the-art machine learning models can be vulnerable to very small input perturbations that are adversarially constructed. Adversarial training is an effective approach to defend against such examples. It is formulated as a min-max problem, searching for the best solution when the training data was corrupted by the worst-case attacks. For linear regression problems, adversarial training can be formulated as a convex problem. We use this reformulation to make two technical contributions: First, we formulate the training problem as an instance of robust regression to reveal its connection to parameter-shrinking methods, specifically that $\ell_\infty$-adversarial training produces sparse solutions. Secondly, we study adversarial training in the overparameterized regime, i.e. when there are more parameters than data. We prove that adversarial training with small disturbances gives the solution with the minimum-norm that interpolates the training data. Ridge regression and lasso approximate such interpolating solutions as their regularization parameter vanishes. By contrast, for adversarial training, the transition into the interpolation regime is abrupt and for non-zero values of disturbance. This result is proved and illustrated with numerical examples.

We study the error of linear regression in the face of adversarial attacks. In this framework, an adversary changes the input to the regression model in order to maximize the prediction error. We provide bounds on the prediction error in the presence of an adversary as a function of the parameter norm and the error in the absence of such an adversary. We show how these bounds make it possible to study the adversarial error using analysis from non-adversarial setups. The obtained results shed light on the robustness of overparameterized linear models to adversarial attacks. Adding features might be either a source of additional robustness or brittleness. On the one hand, we use asymptotic results to illustrate how double-descent curves can be obtained for the adversarial error. On the other hand, we derive conditions under which the adversarial error can grow to infinity as more features are added, while at the same time, the test error goes to zero. We show this behavior is caused by the fact that the norm of the parameter vector grows with the number of features. It is also established that $\ell_\infty$ and $\ell_2$-adversarial attacks might behave fundamentally differently due to how the $\ell_1$ and $\ell_2$-norms of random projections concentrate. We also show how our reformulation allows for solving adversarial training as a convex optimization problem. This fact is then exploited to establish similarities between adversarial training and parameter-shrinking methods and to study how the training might affect the robustness of the estimated models.

The convolutional neural network (CNN) remains an essential tool in solving computer vision problems. Standard convolutional architectures consist of stacked layers of operations that progressively downscale the image. Aliasing is a well-known side-effect of downsampling that may take place: it causes high-frequency components of the original signal to become indistinguishable from its low-frequency components. While downsampling takes place in the max-pooling layers or in the strided-convolutions in these models, there is no explicit mechanism that prevents aliasing from taking place in these layers. Due to the impressive performance of these models, it is natural to suspect that they, somehow, implicitly deal with this distortion. The question we aim to answer in this paper is simply: "how and to what extent do CNNs counteract aliasing?" We explore the question by means of two examples: In the first, we assess the CNNs capability of distinguishing oscillations at the input, showing that the redundancies in the intermediate channels play an important role in succeeding at the task; In the second, we show that an image classifier CNN while, in principle, capable of implementing anti-aliasing filters, does not prevent aliasing from taking place in the intermediate layers., Comment: To appear in the 2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

System identification aims to build models of dynamical systems from data. Traditionally, choosing the model requires the designer to balance between two goals of conflicting nature; the model must be rich enough to capture the system dynamics, but not so flexible that it learns spurious random effects from the dataset. It is typically observed that the model validation performance follows a U-shaped curve as the model complexity increases. Recent developments in machine learning and statistics, however, have observed situations where a "double-descent" curve subsumes this U-shaped model-performance curve. With a second decrease in performance occurring beyond the point where the model has reached the capacity of interpolating - i.e., (near) perfectly fitting - the training data. To the best of our knowledge, such phenomena have not been studied within the context of dynamic systems. The present paper aims to answer the question: "Can such a phenomenon also be observed when estimating parameters of dynamic systems?" We show that the answer is yes, verifying such behavior experimentally both for artificially generated and real-world datasets., Comment: To appear in the Proceedings of the 19th IFAC Symposium in System Identification (2021)

This paper is directed towards the problem of learning nonlinear ARX models based on system input--output data. In particular, our interest is in learning a conditional distribution of the current output based on a finite window of past inputs and outputs. To achieve this, we consider the use of so-called energy-based models, which have been developed in allied fields for learning unknown distributions based on data. This energy-based model relies on a general function to describe the distribution, and here we consider a deep neural network for this purpose. The primary benefit of this approach is that it is capable of learning both simple and highly complex noise models, which we demonstrate on simulated and experimental data.

Recent developments within deep learning are relevant for nonlinear system identification problems. In this paper, we establish connections between the deep learning and the system identification communities. It has recently been shown that convolutional architectures are at least as capable as recurrent architectures when it comes to sequence modeling tasks. Inspired by these results we explore the explicit relationships between the recently proposed temporal convolutional network (TCN) and two classic system identification model structures; Volterra series and block-oriented models. We end the paper with an experimental study where we provide results on two real-world problems, the well-known Silverbox dataset and a newer dataset originating from ground vibration experiments on an F-16 fighter aircraft., Comment: Accepted to Conference on Decision and Control, The first two authors contributed equally

SciPy is an open source scientific computing library for the Python programming language. SciPy 1.0 was released in late 2017, about 16 years after the original version 0.1 release. SciPy has become a de facto standard for leveraging scientific algorithms in the Python programming language, with more than 600 unique code contributors, thousands of dependent packages, over 100,000 dependent repositories, and millions of downloads per year. This includes usage of SciPy in almost half of all machine learning projects on GitHub, and usage by high profile projects including LIGO gravitational wave analysis and creation of the first-ever image of a black hole (M87). The library includes functionality spanning clustering, Fourier transforms, integration, interpolation, file I/O, linear algebra, image processing, orthogonal distance regression, minimization algorithms, signal processing, sparse matrix handling, computational geometry, and statistics. In this work, we provide an overview of the capabilities and development practices of the SciPy library and highlight some recent technical developments., Comment: Article source data is available here: https://github.com/scipy/scipy-articles

The exploding and vanishing gradient problem has been the major conceptual principle behind most architecture and training improvements in recurrent neural networks (RNNs) during the last decade. In this paper, we argue that this principle, while powerful, might need some refinement to explain recent developments. We refine the concept of exploding gradients by reformulating the problem in terms of the cost function smoothness, which gives insight into higher-order derivatives and the existence of regions with many close local minima. We also clarify the distinction between vanishing gradients and the need for the RNN to learn attractors to fully use its expressive power. Through the lens of these refinements, we shed new light on recent developments in the RNN field, namely stable RNN and unitary (or orthogonal) RNNs., Comment: To appear in the Proceedings of the 23rd International Conference on Artificial Intelligence and Statistics (AISTATS), 2020. PMLR: Volume 108. This paper was previously titled "The trade-off between long-term memory and smoothness for recurrent networks". The current version subsumes all previous versions

We shed new light on the \textit{smoothness} of optimization problems arising in prediction error parameter estimation of linear and nonlinear systems. We show that for regions of the parameter space where the model is not contractive, the Lipschitz constant and $\beta$-smoothness of the objective function might blow up exponentially with the simulation length, making it hard to numerically find minima within those regions or, even, to escape from them. In addition to providing theoretical understanding of this problem, this paper also proposes the use of multiple shooting as a viable solution. The proposed method minimizes the error between a prediction model and the observed values. Rather than running the prediction model over the entire dataset, multiple shooting splits the data into smaller subsets and runs the prediction model over each subset, making the simulation length a design parameter and making it possible to solve problems that would be infeasible using a standard approach. The equivalence to the original problem is obtained by including constraints in the optimization. The new method is illustrated by estimating the parameters of nonlinear systems with chaotic or unstable behavior, as well as neural networks. We also present a comparative analysis of the proposed method with multi-step-ahead prediction error minimization.

The role of automatic electrocardiogram (ECG) analysis in clinical practice is limited by the accuracy of existing models. Deep Neural Networks (DNNs) are models composed of stacked transformations that learn tasks by examples. This technology has recently achieved striking success in a variety of task and there are great expectations on how it might improve clinical practice. Here we present a DNN model trained in a dataset with more than 2 million labeled exams analyzed by the Telehealth Network of Minas Gerais and collected under the scope of the CODE (Clinical Outcomes in Digital Electrocardiology) study. The DNN outperform cardiology resident medical doctors in recognizing 6 types of abnormalities in 12-lead ECG recordings, with F1 scores above 80% and specificity over 99%. These results indicate ECG analysis based on DNNs, previously studied in a single-lead setup, generalizes well to 12-lead exams, taking the technology closer to the standard clinical practice., Comment: A preliminary version of this work titled: "Automatic Diagnosis of Short-Duration 12-Lead ECG using a Deep Convolutional Network " was presented in the Machine Learning for Health Workshop at NeurIPS 2018 and was made available under a different identifier: arXiv:1811.12194. The current version subsumes all previous versions

We present a model for predicting electrocardiogram (ECG) abnormalities in short-duration 12-lead ECG signals which outperformed medical doctors on the 4th year of their cardiology residency. Such exams can provide a full evaluation of heart activity and have not been studied in previous end-to-end machine learning papers. Using the database of a large telehealth network, we built a novel dataset with more than 2 million ECG tracings, orders of magnitude larger than those used in previous studies. Moreover, our dataset is more realistic, as it consist of 12-lead ECGs recorded during standard in-clinics exams. Using this data, we trained a residual neural network with 9 convolutional layers to map 7 to 10 second ECG signals to 6 classes of ECG abnormalities. Future work should extend these results to cover a large range of ECG abnormalities, which could improve the accessibility of this diagnostic tool and avoid wrong diagnosis from medical doctors., Comment: Machine Learning for Health (ML4H) Workshop at NeurIPS 2018 arXiv:1811.07216

We propose a new algorithm for estimating NARMAX models with $L_1$ regularization for models represented as a linear combination of basis functions. Due to the $L_1$-norm penalty the Lasso estimation tends to produce some coefficients that are exactly zero and hence gives interpretable models. The novelty of the contribution is the inclusion of error regressors in the Lasso estimation (which yields a nonlinear regression problem). The proposed algorithm uses cyclical coordinate descent to compute the parameters of the NARMAX models for the entire regularization path. It deals with the error terms by updating the regressor matrix along with the parameter vector. In comparative timings we find that the modification does not reduce the computational efficiency of the original algorithm and can provide the most important regressors in very few inexpensive iterations. The method is illustrated for linear and polynomial models by means of two examples., Comment: 2018 American Control Conference

Neural network models for dynamic systems can be trained either in parallel or in series-parallel configurations. Influenced by early arguments, several papers justify the choice of series-parallel rather than parallel configuration claiming it has a lower computational cost, better stability properties during training and provides more accurate results. Other published results, on the other hand, defend parallel training as being more robust and capable of yielding more accu- rate long-term predictions. The main contribution of this paper is to present a study comparing both methods under the same unified framework. We focus on three aspects: i) robustness of the estimation in the presence of noise; ii) computational cost; and, iii) convergence. A unifying mathematical framework and simulation studies show situations where each training method provides better validation results, being parallel training better in what is believed to be more realistic scenarios. An example using measured data seems to reinforce such claim. We also show, with a novel complexity analysis and numerical examples, that both methods have similar computational cost, being series series-parallel training, however, more amenable to parallelization. Some informal discussion about stability and convergence properties is presented and explored in the examples.