Empirical study of the topology and geometry of deep networks

The goal of this paper is to analyze the geometric properties of deep neural network image classifiers in the input space. We specifically study the topology of classification regions created by deep networks, as well as their associated decision boundary. Through a systematic empirical study, we show that state-of-the-art deep nets learn connected classification regions, and that the decision boundary in the vicinity of datapoints is flat along most directions. We further draw an essential connection between two seemingly unrelated properties of deep networks: their sensitivity to additive perturbations of the inputs, and the curvature of their decision boundary. The directions where the decision boundary is curved in fact characterize the directions to which the classifier is the most vulnerable. We finally leverage a fundamental asymmetry in the curvature of the decision boundary of deep nets, and propose a method to discriminate between original images, and images perturbed with small adversarial examples. We show the effectiveness of this purely geometric approach for detecting small adversarial perturbations in images, and for recovering the labels of perturbed images.

Geometry of adversarial robustness of deep networks: methods and applications

Seyed Mohsen Moosavi Dezfooli

We are witnessing a rise in the popularity of using artificial neural networks in many fields of science and technology. Deep neural networks in particular have shown impressive classification performance on a number of challenging benchmarks, generally in well controlled settings. However it is equally important that these classifiers satisfy robustness guarantees when they are deployed in uncontrolled (noise-prone) and possibly hostile environments. In other words, small perturbations applied to the samples should not yield significant loss to the performance of the classifier. Unfortunately, deep neural network classifiers are shown to be intriguingly vulnerable to perturbations and it is relatively easy to design noise that can change the estimated label of the classifier. The study of this high-dimensional phenomenon is a challenging task, and requires the development of new algorithmic tools, as well as theoretical and experimental analysis in order to identify the key factors driving the robustness properties of deep networks. This is exactly the focus of this PhD thesis. First, we propose a computationally efficient yet accurate method to generate minimal perturbations that fool deep neural networks. It permits to reliably quantify the robustness of classifiers and compare different architectures. We further propose a systematic algorithm for computing universal (image-agnostic) and very small perturbation vectors that cause natural images to be misclassified with high probability. The vulnerability to universal perturbations is particularly important in security-critical applications of deep neural networks, and our algorithm shows that these systems are quite vulnerable to noise that is designed with only limited knowledge about test samples or classification architectures. Next, we study the geometry of the classifier's decision boundary in order to explain the adversarial vulnerability of deep networks. Specifically, we establish precise theoretical bounds on the robustness of classifiers in a novel semi-random noise regime that generalizes both the adversarial and the random perturbation regimes. We show in particular that the robustness of deep networks to universal perturbations is driven by a key property of the curvature of their decision boundaries. Finally, we build on the geometric insights derived in this thesis in order to improve the robustness properties of state-of-the-art image classifiers. We leverage a fundamental property in the curvature of the decision boundary of deep networks, and propose a method to detect small adversarial perturbations in images, and to recover the labels of perturbed images. To achieve inherently robust classifiers, we further propose an alternative to the common adversarial training strategy, where we directly minimize the curvature of the classifier. This leads to adversarial robustness that is on par with adversarial training. In summary, we demonstrate in this thesis a new geometric approach to the problem of the adversarial vulnerability of deep networks, and provide novel quantitative and qualitative results that precisely describe the behavior of classifiers in adversarial settings. Our results in this thesis contribute to the understanding of the fundamental properties of state-of-the-art image classifiers that eventually will bring important benefits in safety-critical applications such as in self-driving cars, autonomous robots, and medical imaging.

EPFL2019

Robustness and invariance properties of image classifiers

Apostolos Modas

Deep neural networks have achieved impressive results in many image classification tasks. However, since their performance is usually measured in controlled settings, it is important to ensure that their decisions remain correct when deployed in noisy environments. In fact, deep networks are not robust to a large variety of semantic-preserving image modifications, even to imperceptible image changes -- known as adversarial perturbations -- that can arbitrarily flip the prediction of a classifier. The poor robustness of image classifiers to small data distribution shifts raises serious concerns regarding their trustworthiness. To build reliable machine learning models, we must design principled methods to analyze and understand the mechanisms that shape robustness and invariance. This is exactly the focus of this thesis.First, we study the problem of computing sparse adversarial perturbations, and exploit the geometry of the decision boundaries of image classifiers for computing sparse perturbations very fast. We evaluate the robustness of deep networks to sparse adversarial perturbations in high-dimensional datasets, and reveal a qualitative correlation between the location of the perturbed pixels and the semantic features of the images. Such correlation suggests a deep connection between adversarial examples and the data features that image classifiers learn.To better understand this connection, we provide a geometric framework that connects the distance of data samples to the decision boundary, with the features existing in the data. We show that deep classifiers have a strong inductive bias towards invariance to non-discriminative features, and that adversarial training exploits this property to confer robustness. We demonstrate that the invariances of robust classifiers are useful in data-scarce domains, while the improved understanding of the data influence on the inductive bias of deep networks can be exploited to design more robust classifiers. Finally, we focus on the challenging problem of generalization to unforeseen corruptions of the data, and we propose a novel data augmentation scheme that relies on simple families of max-entropy image transformations to confer robustness to common corruptions. We analyze our method and demonstrate the importance of the mixing strategy on synthesizing corrupted images, and we reveal the robustness-accuracy trade-offs arising in the context of common corruptions. The controllable nature of our method permits to easily adapt it to other tasks and achieve robustness to distribution shifts in data-scarce applications.Overall, our results contribute to the understanding of the fundamental mechanisms of deep image classifiers, and pave the way for building more reliable machine learning systems that can be deployed in real-world environments.

EPFL2022

The Robustness of Deep Networks - A geometric perspective

Alhussein Fawzi, Pascal Frossard, Seyed Mohsen Moosavi Dezfooli

Deep neural networks have recently shown impressive classification performance on a diverse set of visual tasks. When deployed in real-world (noise-prone) environments, it is equally important that these classifiers satisfy robustness guarantees: small perturbations applied to the samples should not yield significant losses to the performance of the predictor. The goal of this paper is to discuss the robustness of deep networks to a diverse set of perturbations that may affect the samples in practice, including adversarial perturbations, random noise, and geometric transformations. Our paper further discusses the recent works that build on the robustness analysis to provide geometric insights on the classifier’s decision surface, which help in developing a better understanding of deep nets. The overview finally presents recent solutions that attempt to increase the robustness of deep networks. We hope that this review paper will contribute shedding light on the open research challenges in the robustness of deep networks, and will stir interest in the analysis of their fundamental properties.