Time-changed normalizing flows for accurate SDE modeling

The generative paradigm has become increasingly important in machine learning and deep learning models. Among popular generative models are normalizing flows, which enable exact likelihood estimation by transforming a base distribution through diffeomorphic transformations. Extending the normalizing flow framework to handle time-indexed flows gave dynamic normalizing flows, a powerful tool to model time series, stochastic processes, and neural stochastic differential equations (SDEs). In this work, we propose a novel variant of dynamic normalizing flows, a Time Changed Normalizing Flow (TCNF), based on time deformation of a Brownian motion which constitutes a versatile and extensive family of Gaussian processes. This approach enables us to effectively model some SDEs, that cannot be modeled otherwise, including standard ones such as the well-known Ornstein-Uhlenbeck process, and generalizes prior methodologies, leading to improved results and better inference and prediction capability.

Measuring and Mitigating Biases in Motor Insurance Pricing

The non-life insurance sector operates within a highly competitive and tightly regulated framework, confronting a pivotal juncture in the formulation of pricing strategies. Insurers are compelled to harness a range of statistical methodologies and available data to construct optimal pricing structures that align with the overarching corporate strategy while accommodating the dynamics of market competition. Given the fundamental societal role played by insurance, premium rates are subject to rigorous scrutiny by regulatory authorities. These rates must conform to principles of transparency, explainability, and ethical considerations. Consequently, the act of pricing transcends mere statistical calculations and carries the weight of strategic and societal factors. These multifaceted concerns may drive insurers to establish equitable premiums, taking into account various variables. For instance, regulations mandate the provision of equitable premiums, considering factors such as policyholder gender or mutualist group dynamics in accordance with respective corporate strategies. Age-based premium fairness is also mandated. In certain insurance domains, variables such as the presence of serious illnesses or disabilities are emerging as new dimensions for evaluating fairness. Regardless of the motivating factor prompting an insurer to adopt fairer pricing strategies for a specific variable, the insurer must possess the capability to define, measure, and ultimately mitigate any ethical biases inherent in its pricing practices while upholding standards of consistency and performance. This study seeks to provide a comprehensive set of tools for these endeavors and assess their effectiveness through practical application in the context of automobile insurance.

Geometry-preserving lie group integrators for differential equations on the manifold of symmetric positive definite matrices

In many applications, one encounters signals that lie on manifolds rather than a Euclidean space. In particular, covariance matrices are examples of ubiquitous mathematical objects that have a non Euclidean structure. The application of Euclidean methods to integrate differential equations lying on such objects does not respect the geometry of the manifold, which can cause many numerical issues. In this paper, we propose to use Lie group methods to define geometry-preserving numerical integration schemes on the manifold of symmetric positive definite matrices. These can be applied to a number of differential equations on covariance matrices of practical interest. We show that they are more stable and robust than other classical or naive integration schemes on an example.

Evaluation of importance estimators in deep learning classifiers for Computed Tomography

Deep learning has shown superb performance in detecting objects and classifying images, ensuring a great promise for analyzing medical imaging. Translating the success of deep learning to medical imaging, in which doctors need to understand the underlying process, requires the capability to interpret and explain the prediction of neural networks. Interpretability of deep neural networks often relies on estimating the importance of input features (e.g., pixels) with respect to the outcome (e.g., class probability). However, a number of importance estimators (also known as saliency maps) have been developed and it is unclear which ones are more relevant for medical imaging applications. In the present work, we investigated the performance of several importance estimators in explaining the classification of computed tomography (CT) images by a convolutional deep network, using three distinct evaluation metrics. First, the model-centric fidelity measures a decrease in the model accuracy when certain inputs are perturbed. Second, concordance between importance scores and the expert-defined segmentation masks is measured on a pixel level by a receiver operating characteristic (ROC) curves. Third, we measure a region-wise overlap between a XRAI-based map and the segmentation mask by Dice Similarity Coefficients (DSC). Overall, two versions of SmoothGrad topped the fidelity and ROC rankings, whereas both Integrated Gradients and SmoothGrad excelled in DSC evaluation. Interestingly, there was a critical discrepancy between model-centric (fidelity) and human-centric (ROC and DSC) evaluation. Expert expectation and intuition embedded in segmentation maps does not necessarily align with how the model arrived at its prediction. Understanding this difference in interpretability would help harnessing the power of deep learning in medicine.

Model Transparency and Interpretability : Survey and Application to the Insurance Industry

The use of models, even if efficient, must be accompanied by an understanding at all levels of the process that transforms data (upstream and downstream). Thus, needs increase to define the relationships between individual data and the choice that an algorithm could make based on its analysis (e.g. the recommendation of one product or one promotional offer, or an insurance rate representative of the risk). Model users must ensure that models do not discriminate and that it is also possible to explain their results. This paper introduces the importance of model interpretation and tackles the notion of model transparency. Within an insurance context, it specifically illustrates how some tools can be used to enforce the control of actuarial models that can nowadays leverage on machine learning. On a simple example of loss frequency estimation in car insurance, we show the interest of some interpretability methods to adapt explanation to the target audience.

Some Remarks on Replicated Simulated Annealing

Recently authors have introduced the idea of training discrete weights neural networks using a mix between classical simulated annealing and a replica ansatz known from the statistical physics literature. Among other points, they claim their method is able to find robust configurations. In this paper, we analyze this so-called "replicated simulated annealing" algorithm. In particular, we explicit criteria to guarantee its convergence, and study when it successfully samples from configurations. We also perform experiments using synthetic and real data bases.

Interpretabilité des modèles : état des lieux des méthodes et application à l'assurance

Since May 2018, the General Data Protection Regulation (GDPR) has introduced new obligations to industries. By setting a legal framework, it notably imposes strong transparency on the use of personal data. Thus, people must be informed of the use of their data and must consent the usage of it. Data is the raw material of many models which today make it possible to increase the quality and performance of digital services. Transparency on the use of data also requires a good understanding of its use through different models. The use of models, even if efficient, must be accompanied by an understanding at all levels of the process that transform data (upstream and downstream of a model), thus making it possible to define the relationships between the individual's data and the choice that an algorithm could make based on the analysis of the latter. (For example, the recommendation of one product or one promotional offer or an insurance rate representative of the risk.) Models users must ensure that models do not discriminate against and that it is also possible to explain its result. The widening of the panel of predictive algorithms - made possible by the evolution of computing capacities -- leads scientists to be vigilant about the use of models and to consider new tools to better understand the decisions deduced from them . Recently, the community has been particularly active on model transparency with a marked intensification of publications over the past three years. The increasingly frequent use of more complex algorithms (\textit{deep learning}, Xgboost, etc.) presenting attractive performances is undoubtedly one of the causes of this interest. This article thus presents an inventory of methods of interpreting models and their uses in an insurance context.

Towards an Intrinsic Definition of Robustness for a Classifier

The robustness of classifiers has become a question of paramount importance in the past few years. Indeed, it has been shown that state-of-the-art deep learning architectures can easily be fooled with imperceptible changes to their inputs. Therefore, finding good measures of robustness of a trained classifier is a key issue in the field. In this paper, we point out that averaging the radius of robustness of samples in a validation set is a statistically weak measure. We propose instead to weight the importance of samples depending on their difficulty. We motivate the proposed score by a theoretical case study using logistic regression, where we show that the proposed score is independent of the choice of the samples it is evaluated upon. We also empirically demonstrate the ability of the proposed score to measure robustness of classifiers with little dependence on the choice of samples in more complex settings, including deep convolutional neural networks and real datasets.

Improving Accuracy of Nonparametric Transfer Learning via Vector Segmentation

Transfer learning using deep neural networks as feature extractors has become increasingly popular over the past few years. It allows to obtain state-of-the-art accuracy on datasets too small to train a deep neural network on its own, and it provides cutting edge descriptors that, combined with nonparametric learning methods, allow rapid and flexible deployment of performing solutions in computationally restricted settings. In this paper, we are interested in showing that the features extracted using deep neural networks have specific properties which can be used to improve accuracy of downstream nonparametric learning methods. Namely, we demonstrate that for some distributions where information is embedded in a few coordinates, segmenting feature vectors can lead to better accuracy. We show how this model can be applied to real datasets by performing experiments using three mainstream deep neural network feature extractors and four databases, in vision and audio.