Deep learning models have been the subject of study from various perspectives, for example, their training process, interpretation, generalization error, robustness to adversarial attacks, etc. A trained model is defined by its decision boundaries, and therefore, many of the studies about deep learning models speculate about the decision boundaries, and sometimes make simplifying assumptions about them. So far, finding exact points on the decision boundaries of trained deep models has been considered an intractable problem. Here, we compute exact points on the decision boundaries of these models and provide mathematical tools to investigate the surfaces that define the decision boundaries. Through numerical results, we confirm that some of the speculations about the decision boundaries are accurate, some of the computational methods can be improved, and some of the simplifying assumptions may be unreliable, for models with nonlinear activation functions. We advocate for verification of simplifying assumptions and approximation methods, wherever they are used. Finally, we demonstrate that the computational practices used for finding adversarial examples can be improved and computing the closest point on the decision boundary reveals the weakest vulnerability of a model against adversarial attack.
Fraunhofer IAIS（弗劳恩霍夫智能分析与信息系统研究所）；ML2R（机器学习研究中心）；University of Cologne（科隆大学）；IIT Bombay（印度理工学院孟买分校）
In the present work we study classifiers' decision boundaries via Brownian motion processes in ambient data space and associated probabilistic techniques. Intuitively, our ideas correspond to placing a heat source at the decision boundary and observing how effectively the sample points warm up. We are largely motivated by the search for a soft measure that sheds further light on the decision boundary's geometry. En route, we bridge aspects of potential theory and geometric analysis (Mazya, 2011, Grigoryan-Saloff-Coste, 2002) with active fields of ML research such as adversarial examples and ge…
University of Vienna（维也纳大学）；Katholische Universität Eichstätt-Ingolstadt（艾希施泰特-因戈尔施塔特天主教大学）；Technical University of Munich（慕尼黑工业大学）
We study the problem of learning classification functions from noiseless training samples, under the assumption that the decision boundary is of a certain regularity. We establish universal lower bounds for this estimation problem, for general classes of continuous decision boundaries. For the class of locally Barron-regular decision boundaries, we find that the optimal estimation rates are essentially independent of the underlying dimension and can be realized by empirical risk minimization methods over a suitable class of deep neural networks. These results are based on novel estimates of th…
Geometry and topology of decision regions are closely related with classification performance and robustness against adversarial attacks. In this paper, we use differential geometry to theoretically explore the geometrical and topological properties of decision regions produced by deep neural networks (DNNs). The goal is to obtain some geometrical and topological properties of decision boundaries for given DNN models, and provide some principled guidance to design and regularization of DNNs. First, we present the curvatures of decision boundaries in terms of network parameters, and give suffic…
King Abdullah University of Science and Technology (KAUST)（阿卜杜拉国王科技大学）；University of Oxford（牛津大学）；Zalando SE（Zalando公司）；itemis AG（itemis股份公司）
This work tackles the problem of characterizing and understanding the decision boundaries of neural networks with piecewise linear non-linearity activations. We use tropical geometry, a new development in the area of algebraic geometry, to characterize the decision boundaries of a simple network of the form (Affine, ReLU, Affine). Our main finding is that the decision boundaries are a subset of a tropical hypersurface, which is intimately related to a polytope formed by the convex hull of two zonotopes. The generators of these zonotopes are functions of the network parameters. This geometric c…
We propose the labeled Čech complex, the plain labeled Vietoris-Rips complex, and the locally scaled labeled Vietoris-Rips complex to perform persistent homology inference of decision boundaries in classification tasks. We provide theoretical conditions and analysis for recovering the homology of a decision boundary from samples. Our main objective is quantification of deep neural network complexity to enable matching of datasets to pre-trained models; we report results for experiments using MNIST, FashionMNIST, and CIFAR10.
University of Maryland, College Park（马里兰大学帕克分校）；Rice University（莱斯大学）；New York University（纽约大学）
We discuss methods for visualizing neural network decision boundaries and decision regions. We use these visualizations to investigate issues related to reproducibility and generalization in neural network training. We observe that changes in model architecture (and its associate inductive bias) cause visible changes in decision boundaries, while multiple runs with the same architecture yield results with strong similarities, especially in the case of wide architectures. We also use decision boundary methods to visualize double descent phenomena. We see that decision boundary reproducibility d…
Tsinghua University（清华大学）；Institute for Interdisciplinary Information Sciences（交叉信息研究院）
Is it possible to train several classifiers to perform meaningful crowd-sourcing to produce a better prediction label set without ground-truth annotation? This paper will modify the contrastive learning objectives to automatically train a self-complementing ensemble to produce a state-of-the-art prediction on the CIFAR10 and CIFAR100-20 tasks. This paper will present a straightforward method to modify a single unsupervised classification pipeline to automatically generate an ensemble of neural networks with varied decision boundaries to learn a more extensive feature set of classes. Loss Funct…
Centre Borelli（博雷利中心）；ENS Paris-Saclay（巴黎萨克雷高等师范学院）
When analyzing a dataset, it can be useful to assess how smooth the decision boundaries need to be for a model to better fit the data. This paper addresses this question by proposing the quantification of how much should the 'rigid' decision boundaries, produced by an algorithm that naturally finds such solutions, be relaxed to obtain a performance improvement. The approach we propose starts with the rigid decision boundaries of a seed Decision Tree (seed DT), which is used to initialize a Neural DT (NDT). The initial boundaries are challenged by relaxing them progressively through training th…
Trilateral Research（三边研究公司）；Data Justice Lab, Cardiff University（卡迪夫大学数据正义实验室）；University of Granada（格拉纳达大学）
Fair machine learning works have been focusing on the development of equitable algorithms that address discrimination of certain groups. Yet, many of these fairness-aware approaches aim to obtain a unique solution to the problem, which leads to a poor understanding of the statistical limits of bias mitigation interventions. We present the first methodology that allows to explore those limits within a multi-objective framework that seeks to optimize any measure of accuracy and fairness and provides a Pareto front with the best feasible solutions. In this work, we focus our study on decision tre…
We consider the recent privacy preserving methods that train the models not on original images, but on mixed images that look like noise and hard to trace back to the original images. We explain that those mixed images will be samples on the decision boundaries of the trained model, and although such methods successfully hide the contents of images from the entity in charge of federated learning, they provide crucial information to that entity about the decision boundaries of the trained model. Once the entity has exact samples on the decision boundaries of the model, they may use it for effec…
Zhejiang University of Technology（浙江工业大学）；Zhejiang University（浙江大学）
With the successful application of deep learning models in many real-world tasks, the model robustness becomes more and more critical. Often, we evaluate the robustness of the deep models by attacking them with purposely generated adversarial samples, which is computationally costly and dependent on the specific attackers and the model types. This work proposes a generic evaluation metric ROBY, a novel attack-independent robustness measure based on the model's decision boundaries. Independent of adversarial samples, ROBY uses the inter-class and intra-class statistic features to capture the fe…
Arizona State University（亚利桑那州立大学）；IBM Research（IBM研究院）
Accurately and efficiently characterizing the decision boundary of classifiers is important for problems related to model selection and meta-learning. Inspired by topological data analysis, the characterization of decision boundaries using their homology has recently emerged as a general and powerful tool. In this paper, we propose an active learning algorithm to recover the homology of decision boundaries. Our algorithm sequentially and adaptively selects which samples it requires the labels of. We theoretically analyze the proposed framework and show that the query complexity of our active l…
Centre Borelli（博雷利中心）；ENS Paris-Saclay（巴黎萨克雷高等师范学院）；Universite Paris-Saclay（巴黎萨克雷大学）
Model selection consists in comparing several candidate models according to a metric to be optimized. The process often involves a grid search, or such, and cross-validation, which can be time consuming, as well as not providing much information about the dataset itself. In this paper we propose a method to reduce the scope of exploration needed for the task. The idea is to quantify how much it would be necessary to depart from trained instances of a given family, reference models (RMs) carrying `rigid' decision boundaries (e.g. decision trees), so as to obtain an equivalent or better model. I…
Capacity control, the bias/variance dilemma, and learning unknown functions from data, are all concerned with identifying effective and consistent fits of unknown geometric loci to random data points. A geometric locus is a curve or surface formed by points, all of which possess some uniform property. A geometric locus of an algebraic equation is the set of points whose coordinates are solutions of the equation. Any given curve or surface must pass through each point on a specified locus. This paper argues that it is impossible to fit random data points to algebraic equations of partially conf…
Important insights towards the explainability of neural networks reside in the characteristics of their decision boundaries. In this work, we borrow tools from the field of adversarial robustness, and propose a new perspective that relates dataset features to the distance of samples to the decision boundary. This enables us to carefully tweak the position of the training samples and measure the induced changes on the boundaries of CNNs trained on large-scale vision datasets. We use this framework to reveal some intriguing properties of CNNs. Specifically, we rigorously confirm that neural netw…
Yale University（耶鲁大学）；University of Maryland, College Park（马里兰大学帕克分校）
Deep learning models have been criticized for their lack of easy interpretation, which undermines confidence in their use for important applications. Nevertheless, they are consistently utilized in many applications, consequential to humans' lives, mostly because of their better performance. Therefore, there is a great need for computational methods that can explain, audit, and debug such models. Here, we use flip points to accomplish these goals for deep learning models with continuous output scores (e.g., computed by softmax), used in social applications. A flip point is any point that lies …
Fraunhofer Institute for Experimental Software Engineering IESE（弗劳恩霍夫实验软件工程研究所）
Outcomes of data-driven AI models cannot be assumed to be always correct. To estimate the uncertainty in these outcomes, the uncertainty wrapper framework has been proposed, which considers uncertainties related to model fit, input quality, and scope compliance. Uncertainty wrappers use a decision tree approach to cluster input quality related uncertainties, assigning inputs strictly to distinct uncertainty clusters. Hence, a slight variation in only one feature may lead to a cluster assignment with a significantly different uncertainty. Our objective is to replace this with an approach that m…
Graz University of Technology（格拉茨工业大学）；Medical University Graz（格拉茨医科大学）
In this paper we propose a new augmentation technique, called patch augmentation, that, in our experiments, improves model accuracy and makes networks more robust to adversarial attacks. In brief, this data-independent approach creates new image data based on image/label pairs, where a patch from one of the two images in the pair is superimposed on to the other image, creating a new augmented sample. The new image's label is a linear combination of the image pair's corresponding labels. Initial experiments show a several percentage point increase in accuracy on CIFAR-10, from a baseline of app…
The notion of concept drift refers to the phenomenon that the distribution generating the observed data changes over time. If drift is present, machine learning models may become inaccurate and need adjustment. Many technologies for learning with drift rely on the interleaved test-train error (ITTE) as a quantity which approximates the model generalization error and triggers drift detection and model updates. In this work, we investigate in how far this procedure is mathematically justified. More precisely, we relate a change of the ITTE to the presence of real drift, i.e., a changed posterior…
Machine learning algorithms using deep architectures have been able to implement increasingly powerful and successful models. However, they also become increasingly more complex, more difficult to comprehend and easier to fool. So far, most methods in the literature investigate the decision of the model for a single given input datum. In this paper, we propose to visualize a part of the decision function of a deep neural network together with a part of the data set in two dimensions with discriminative dimensionality reduction. This enables us to inspect how different properties of the data ar…
We study the generalization of deep learning models in relation to the convex hull of their training sets. A trained image classifier basically partitions its domain via decision boundaries and assigns a class to each of those partitions. The location of decision boundaries inside the convex hull of training set can be investigated in relation to the training samples. However, our analysis shows that in standard image classification datasets, all testing images are considerably outside that convex hull, in the pixel space, in the wavelet space, and in the internal representations learned by de…
We show that unlike machine learning classifiers, there are no complex boundary structures in the decision boundaries for well-trained deep models. However, we found that the complicated structures do appear in training but they vanish shortly after shaping. This is a pessimistic news if one seeks to capture different levels of complexity in the decision boundary for understanding generalization, which works well in machine learning. Nonetheless, we found that the decision boundaries of predecessor models on the training data are reflective of the final model's generalization. We show how to u…
Yale Center for Medical Informatics（耶鲁医学信息学中心）；VA CT Healthcare System（康涅狄格州退伍军人医疗保健系统）
The success of deep neural networks in image classification and learning can be partly attributed to the features they extract from images. It is often speculated about the properties of a low-dimensional manifold that models extract and learn from images. However, there is not sufficient understanding about this low-dimensional space based on theory or empirical evidence. For image classification models, their last hidden layer is the one where images of each class is separated from other classes and it also has the least number of features. Here, we develop methods and formulations to study …
Stanford University（斯坦福大学）；University of California, Los Angeles（加州大学洛杉矶分校）；New York University（纽约大学）
With the rise of the digital economy and an explosion of available information about consumers, effective personalization of goods and services has become a core business focus for companies to improve revenues and maintain a competitive edge. This paper studies the personalization problem through the lens of policy learning, where the goal is to learn a decision-making rule (a policy) that maps from consumer and product characteristics (features) to recommendations (actions) in order to optimize outcomes (rewards). We focus on using available historical data for offline learning with unknown …
Machine learning models may involve decision boundaries that change over time due to updates to rules and regulations, such as in loan approvals or claims management. However, in such scenarios, it may take time for sufficient training data to accumulate in order to retrain the model to reflect the new decision boundaries. While work has been done to reinforce existing decision boundaries, very little has been done to cover these scenarios where decision boundaries of the ML models should change in order to reflect new rules. In this paper, we focus on user-provided feedback rules as a way to …
Deep neural networks and in particular, deep neural classifiers have become an integral part of many modern applications. Despite their practical success, we still have limited knowledge of how they work and the demand for such an understanding is evergrowing. In this regard, one crucial aspect of deep neural network classifiers that can help us deepen our knowledge about their decision-making behavior is to investigate their decision boundaries. Nevertheless, this is contingent upon having access to samples populating the areas near the decision boundary. To achieve this, we propose a novel a…
Pittsburgh, PA 15213, USA（美国宾夕法尼亚州匹兹堡（注：该地址不属于机构，可能是原数据错误，按要求应忽略，此处仅为示例说明））
Recent work has found that adversarially-robust deep networks used for image classification are more interpretable: their feature attributions tend to be sharper, and are more concentrated on the objects associated with the image's ground-truth class. We show that smooth decision boundaries play an important role in this enhanced interpretability, as the model's input gradients around data points will more closely align with boundaries' normal vectors when they are smooth. Thus, because robust models have smoother boundaries, the results of gradient-based attribution methods, like Integrated G…
University of Pretoria（比勒陀利亚大学）；Council for Scientific and Industrial Research（科学与工业研究理事会）；Government of Sierra Leone（塞拉利昂政府）
Available or adequate information to inform decision making for resource allocation in support of school improvement is a critical issue globally. In this paper, we apply machine learning and education data mining techniques on education big data to identify determinants of high schools' performance in two African countries: South Africa and Sierra Leone. The research objective is to build predictors for school performance and extract the importance of different community and school-level features. We deploy interpretable metrics from machine learning approaches such as SHAP values on tree mod…
Azad University, Science and Research Branch（阿扎德大学科学与研究分校）；University of California, San Diego（加利福尼亚大学圣迭戈分校）
In medical science, it is very important to gather multiple data on different diseases and one of the most important objectives of the data is to investigate the diseases. Myocardial infarction is a serious risk factor in mortality and in previous studies, the main emphasis has been on people with heart disease and measuring the likelihood of myocardial infarction in them through demographic features, echocardiography, and electrocardiogram. In contrast, the purpose of the present study is to utilize data analysis algorithms and compare their accuracy in patients with a heart attack in order t…
A useful method for representing Bayesian classifiers is through \emph{discriminant functions}. Here, using copula functions, we propose a new model for discriminants. This model provides a rich and generalized class of decision boundaries. These decision boundaries significantly boost the classification accuracy especially for high dimensional feature spaces. We strengthen our analysis through simulation results.
University of Maryland（马里兰大学）；MPI-SWS（马克斯·普朗克软件系统研究所）
Deep neural networks (DNNs) are increasingly used in real-world applications (e.g. facial recognition). This has resulted in concerns about the fairness of decisions made by these models. Various notions and measures of fairness have been proposed to ensure that a decision-making system does not disproportionately harm (or benefit) particular subgroups of the population. In this paper, we argue that traditional notions of fairness that are only based on models' outputs are not sufficient when the model is vulnerable to adversarial attacks. We argue that in some cases, it may be easier for an a…
Pushing the (Decision) Boundaries: Dynamically Calibrating Differentially Private Noise to Explainability in Federated Learning
School of Industrial & Intelligent Systems Engineering, Tel Aviv University（特拉维夫大学工业与智能系统工程学院）；School of Integrated Technology, Yonsei University（延世大学融合技术学院）
Investigating disaster response through social media data and the Susceptible-Infected-Recovered (SIR) model: A case study of 2020 Western U.S. wildfire season
University of Maryland（马里兰大学）；University of Michigan（密歇根大学）；Pennsylvania State University（宾夕法尼亚州立大学）
The research process includes many decisions, e.g., how to entitle and where to publish the paper. In this paper, we introduce a general framework for investigating the effects of such decisions. The main difficulty in investigating the effects is that we need to know counterfactual results, which are not available in reality. The key insight of our framework is inspired by the existing counterfactual analysis using twins, where the researchers regard twins as counterfactual units. The proposed framework regards a pair of papers that cite each other as twins. Such papers tend to be parallel wo…
