acase@tulane.edu / The technological advances of past decades have led to the accumulation of large amounts of genomic and neuroimaging data, enabling novel strategies in precision medicine. These largely rely on machine learning algorithms and modern statistical methods for big biological datasets, which are data-driven rather than hypothesis-driven. These methods often lack guarantees on the validity of the research findings. Because it can be a matter of life and death, when computational methods are deployed in clinical practice in medicine, establishing guarantees on the validity of the results is essential for the advancement of precision medicine. This thesis proposes several novel sparse regression and sparse canonical correlation analysis techniques, which by design include guarantees on the false discovery rate in variable selection. Variable selection on biomedical data is essential for many areas of healthcare, including precision medicine, population stratification, drug development, and predictive modeling of disease phenotypes. Predictive machine learning models can directly affect the patient when used to aid diagnosis, and therefore they need to be thoroughly evaluated before deployment. We present a novel approach to validly reuse the test data for performance evaluation of predictive models. The proposed methods are validated in the application on large genomic and neuroimaging datasets, where they confirm results from previous studies and also lead to new biological insights. In addition, this work puts a focus on making the proposed methods widely available to the scientific community though the release of free and open-source scientific software. / 1 / Alexej Gossmann

Master of Science / Department of Statistics / Kun Chen / We study the composite group bridge penalized regression methods for conducting bilevel variable selection in high dimensional linear regression models with a diverging number of predictors. The proposed method combines the ideas of bridge regression (Huang et al., 2008a) and group bridge regression (Huang et al., 2009), to achieve variable selection consistency in both individual and group levels simultaneously, i.e., the important groups and the important individual variables within each group can both be correctly identi ed with probability approaching to one as the sample size increases to in nity. The method takes full advantage of the prior grouping information, and the established bi-level oracle properties ensure that the method is immune to possible group misidenti cation. A related adaptive group bridge estimator, which uses adaptive penalization for improving bi-level selection, is also investigated. Simulation studies show that the proposed methods have superior performance in comparison to many existing methods.

abstract: Imaging genetics is an emerging and promising technique that investigates how genetic variations affect brain development, structure, and function. By exploiting disorder-related neuroimaging phenotypes, this class of studies provides a novel direction to reveal and understand the complex genetic mechanisms. Oftentimes, imaging genetics studies are challenging due to the relatively small number of subjects but extremely high-dimensionality of both imaging data and genomic data. In this dissertation, I carry on my research on imaging genetics with particular focuses on two tasks---building predictive models between neuroimaging data and genomic data, and identifying disorder-related genetic risk factors through image-based biomarkers. To this end, I consider a suite of structured sparse methods---that can produce interpretable models and are robust to overfitting---for imaging genetics. With carefully-designed sparse-inducing regularizers, different biological priors are incorporated into learning models. More specifically, in the Allen brain image--gene expression study, I adopt an advanced sparse coding approach for image feature extraction and employ a multi-task learning approach for multi-class annotation. Moreover, I propose a label structured-based two-stage learning framework, which utilizes the hierarchical structure among labels, for multi-label annotation. In the Alzheimer's disease neuroimaging initiative (ADNI) imaging genetics study, I employ Lasso together with EDPP (enhanced dual polytope projections) screening rules to fast identify Alzheimer's disease risk SNPs. I also adopt the tree-structured group Lasso with MLFre (multi-layer feature reduction) screening rules to incorporate linkage disequilibrium information into modeling. Moreover, I propose a novel absolute fused Lasso model for ADNI imaging genetics. This method utilizes SNP spatial structure and is robust to the choice of reference alleles of genotype coding. In addition, I propose a two-level structured sparse model that incorporates gene-level networks through a graph penalty into SNP-level model construction. Lastly, I explore a convolutional neural network approach for accurate predicting Alzheimer's disease related imaging phenotypes. Experimental results on real-world imaging genetics applications demonstrate the efficiency and effectiveness of the proposed structured sparse methods. / Dissertation/Thesis / Doctoral Dissertation Computer Science 2017

Sparse regressions based on constraining the L1-norm of the coefficients became popular due to their ability to handle high dimensional data unlike the regular regressions which suffer from overfitting and model identifiability issues especially when sample size is small. They are often the method of choice in many fields of science and engineering for simultaneously selecting covariates and fitting parsimonious linear models that are better generalizable and easily interpretable. However, significant challenges may be posed by the need to accommodate extremes and other domain constraints such as dynamical relations among variables, spatial and temporal constraints, need to provide uncertainty estimates and feature correlations, among others. We adopted a hierarchical Bayesian version of the sparse regression framework and exploited its inherent flexibility to accommodate the constraints. We applied sparse regression for the feature selection problem of statistical downscaling of the climate variables with particular focus on their extremes. This is important for many impact studies where the climate change information is required at a spatial scale much finer than that provided by the global or regional climate models. Characterizing the dependence of extremes on covariates can help in identification of plausible causal drivers and inform extremes downscaling. We propose a general-purpose sparse Bayesian framework for covariate discovery that accommodates the non-Gaussian distribution of extremes within a hierarchical Bayesian sparse regression model. We obtain posteriors over regression coefficients, which indicate dependence of extremes on the corresponding covariates and provide uncertainty estimates, using a variational Bayes approximation. The method is applied for selecting informative atmospheric covariates at multiple spatial scales as well as indices of large scale circulation and global warming related to frequency of precipitation extremes over continental United States. Our results confirm the dependence relations that may be expected from known precipitation physics and generates novel insights which can inform physical understanding. We plan to extend our model to discover covariates for extreme intensity in future. We further extend our framework to handle the dynamic relationship among the climate variables using a nonparametric Bayesian mixture of sparse regression models based on Dirichlet Process (DP). The extended model can achieve simultaneous clustering and discovery of covariates within each cluster. Moreover, the a priori knowledge about association between pairs of data-points is incorporated in the model through must-link constraints on a Markov Random Field (MRF) prior. A scalable and efficient variational Bayes approach is developed to infer posteriors on regression coefficients and cluster variables. / Computer and Information Science

Le thème général de cette thèse est celui de la construction de modèles permettantd’approximer une fonction f lorsque la valeur de f(x) est connue pour un certainnombre de points x. Les modèles considérés ici, souvent appelés modèles de krigeage,peuvent être abordés suivant deux points de vue : celui de l’approximation dans les espacesde Hilbert à noyaux reproduisants ou celui du conditionnement de processus gaussiens.Lorsque l’on souhaite modéliser une fonction dépendant d’une dizaine de variables, lenombre de points nécessaires pour la construction du modèle devient très important etles modèles obtenus sont difficilement interprétables. A partir de ce constat, nous avonscherché à construire des modèles simplifié en travaillant sur un objet clef des modèles dekrigeage : le noyau. Plus précisement, les approches suivantes sont étudiées : l’utilisation denoyaux additifs pour la construction de modèles additifs et la décomposition des noyauxusuels en sous-noyaux pour la construction de modèles parcimonieux. Pour finir, nousproposons une classe de noyaux qui est naturellement adaptée à la représentation ANOVAdes modèles associés et à l’analyse de sensibilité globale. / The framework of this thesis is the approximation of functions for which thevalue is known at limited number of points. More precisely, we consider here the so-calledkriging models from two points of view : the approximation in reproducing kernel Hilbertspaces and the Gaussian Process regression.When the function to approximate depends on many variables, the required numberof points can become very large and the interpretation of the obtained models remainsdifficult because the model is still a high-dimensional function. In light of those remarks,the main part of our work adresses the issue of simplified models by studying a key conceptof kriging models, the kernel. More precisely, the following aspects are adressed: additivekernels for additive models and kernel decomposition for sparse modeling. Finally, wepropose a class of kernels that is well suited for functional ANOVA representation andglobal sensitivity analysis.

Le déploiement généralisé de modèles d’apprentissage automatique de plus en plus performants a entraîné des pressions croissantes pour améliorer la robustesse, la sécurité et l’équité de ces modèles—-souvent en raison de considérations réglementaires et éthiques. En outre, la mise en œuvre de solutions d’intelligence artificielle dans des applications réelles est limitée par leur incapacité actuelle à garantir la conformité aux normes industrielles et aux réglementations gouvernementales. Les pipelines standards pour le développement de modèles d’apprentissage automatique adoptent une mentalité de “construire maintenant, réparer plus tard”, intégrant des mesures de sécurité a posteriori. Cette accumulation continue de dette technique entrave le progrès du domaine à long terme. L’optimisation sous contraintes offre un cadre conceptuel accompagné d’outils algorithmiques permettant d’imposer de manière fiable des propriétés complexes sur des modèles d’apprentissage automatique. Cette thèse appelle à un changement de paradigme dans lequel les contraintes constituent une partie intégrante du processus de développement des modèles, visant à produire des modèles d’apprentissage automatique qui sont intrinsèquement sécurisés par conception. Cette thèse offre une perspective holistique sur l’usage de l’optimisation sous contraintes dans les tâches d’apprentissage profond. Nous examinerons i) la nécessité de formulations contraintes, ii) les avantages offerts par le point de vue de l’optimisation sous contraintes, et iii) les défis algorithmiques qui surgissent dans la résolution de ces problèmes. Nous présentons plusieurs études de cas illustrant l’application des techniques d’optimisation sous contraintes à des problèmes courants d’apprentissage automatique. Dans la Contribution I, nous plaidons en faveur de l’utilisation des formulations sous contraintes en apprentissage automatique. Nous soutenons qu’il est préférable de gérer des régularisateurs interprétables via des contraintes explicites plutôt que par des pénalités additives, particulièrement lorsqu’il s’agit de modèles non convexes. Nous considérons l’entraînement de modèles creux avec une régularisation L0 et démontrons que i) il est possible de trouver des solutions réalisables et performantes à des problèmes de grande envergure avec des contraintes non convexes ; et que ii) l’approche contrainte peut éviter les coûteux ajustements par essais et erreurs inhérents aux techniques basées sur les pénalités. La Contribution II approfondit la contribution précédente en imposant des contraintes explicites sur le taux de compression atteint par les Représentations Neuronales Implicites—-une classe de modèles visant à entreposer efficacement des données (telles qu’une image) dans les paramètres d’un réseau neuronal. Dans ce travail, nous nous concentrons sur l’interaction entre la taille du modèle, sa capacité représentationnelle, et le temps d’entraînement requis. Plutôt que de restreindre la taille du modèle à un budget fixe (qui se conforme au taux de compression requis), nous entraînons un modèle surparamétré et creux avec des contraintes de taux de compression. Cela nous permet d’exploiter la puissance de modèles plus grands pour obtenir de meilleures reconstructions, plus rapidement, sans avoir à nous engager à leur taux de compression indésirable. La Contribution III présente les avantages des formulations sous contraintes dans une application réaliste de la parcimonie des modèles avec des contraintes liées à l’équité non différentiables. Les performances des réseaux neuronaux élagués se dégradent de manière inégale entre les sous-groupes de données, nécessitant ainsi l’utilisation de techniques d’atténuation. Nous proposons une formulation qui impose des contraintes sur les changements de précision du modèle dans chaque sous-groupe, contrairement aux travaux antérieurs qui considèrent des contraintes basées sur des métriques de substitution (telles que la perte du sous-groupe). Nous abordons les défis de la non-différentiabilité et de la stochasticité posés par nos contraintes proposées, et démontrons que notre méthode s’adapte de manière fiable aux problèmes d’optimisation impliquant de grands modèles et des centaines de sous-groupes. Dans la Contribution IV, nous nous concentrons sur la dynamique de l’optimisation lagrangienne basée sur le gradient, une technique populaire pour résoudre les problèmes sous contraintes non convexes en apprentissage profond. La nature adversariale du jeu min-max lagrangien le rend sujet à des comportements oscillatoires ou instables. En nous basant sur des idées tirées de la littérature sur les régulateurs PID, nous proposons un algorithme pour modifier les multiplicateurs de Lagrange qui offre une dynamique d’entraînement robuste et stable. Cette contribution met en place les bases pour que les praticiens adoptent et mettent en œuvre des approches sous contraintes avec confiance dans diverses applications réelles. Dans la Contribution V, nous fournissons un aperçu de Cooper : une bibliothèque pour l’optimisation sous contraintes basée sur le lagrangien dans PyTorch. Cette bibliothèque open-source implémente toutes les contributions principales présentées dans les chapitres précédents et s’intègre harmonieusement dans le cadre PyTorch. Nous avons développé Cooper dans le but de rendre les techniques d’optimisation sous contraintes facilement accessibles aux chercheurs et praticiens de l’apprentissage automatique. / The widespread deployment of increasingly capable machine learning models has resulted in mounting pressures to enhance the robustness, safety and fairness of such models--often arising from regulatory and ethical considerations. Further, the implementation of artificial intelligence solutions in real-world applications is limited by their current inability to guarantee compliance with industry standards and governmental regulations. Current standard pipelines for developing machine learning models embrace a “build now, fix later” mentality, retrofitting safety measures as afterthoughts. This continuous incurrence of technical debt hinders the progress of the field in the long-term. Constrained optimization offers a conceptual framework accompanied by algorithmic tools for reliably enforcing complex properties on machine learning models. This thesis calls for a paradigm shift in which constraints constitute an integral part of the model development process, aiming to produce machine learning models that are inherently secure by design. This thesis provides a holistic perspective on the use of constrained optimization in deep learning tasks. We shall explore i) the need for constrained formulations, ii) the advantages afforded by the constrained optimization standpoint and iii) the algorithmic challenges arising in the solution of such problems. We present several case-studies illustrating the application of constrained optimization techniques to popular machine learning problems. In Contribution I, we advocate for the use of constrained formulations in machine learning. We argue that it is preferable to handle interpretable regularizers via explicit constraints, rather than using additive penalties, specially when dealing with non-convex models. We consider the training of sparse models with L0-regularization and demonstrate that i) it is possible to find feasible, well-performing solutions to large-scale problems with non-convex constraints; and that ii) the constrained approach can avoid the costly trial-and-error tuning inherent to penalty-based techniques. Contribution II expands on the previous contribution by imposing explicit constraints on the compression-rate achieved by Implicit Neural Representations—-a class of models that aim to efficiently store data (such as an image) within a neural network’s parameters. In this work we concentrate on the interplay between the model size, its representational capacity and the required training time. Rather than restricting the model size to a fixed budget (that complies with the required compression rate), we train an overparametrized, sparse model with compression-rate constraints. This allows us to exploit the power of larger models to achieve better reconstructions, faster; without having to commit to their undesirable compression rate. Contribution III showcases the advantages of constrained formulations in a realistic model sparsity application with non-differentiable fairness-related constraints. The performance of pruned neural networks degrades unevenly across data sub-groups, thus requiring the use of mitigation techniques. We propose a formulation that imposes constraints on changes in the model accuracy in each sub-group, in contrast to prior work which considers constraints based on surrogate metrics (such as the sub-group loss). We address the non-differentiability and stochasticity challenges posed by our proposed constraints, and demonstrate that our method scales reliably to optimization problems involving large models and hundreds of sub-groups. In Contribution IV, we focus on the dynamics of gradient-based Lagrangian optimization, a popular technique for solving the non-convex constrained problems arising in deep learning. The adversarial nature of the min-max Lagrangian game makes it prone to oscillatory or unstable behaviors. Based on ideas from the PID control literature, we propose an algorithm for updating the Lagrange multipliers which yields robust, stable training dynamics. This contribution lays the groundwork for practitioners to adopt and implement constrained approaches confidently in diverse real-world applications. In Contribution V, we provide an overview of Cooper: a library for Lagrangian-based constrained optimization in PyTorch. This open-source library implements all the core contributions presented in the preceding chapters and integrates seamlessly with the PyTorch framework. We developed Cooper with the goal of making constrained optimization techniques readily available to machine learning researchers and practitioners.