A surprising phenomenon in modern machine learning is the ability of a highly overparameterized model to generalize well (small error on the test data) even when it is trained to memorize the training data (zero error on the training data). This has led to an arms race towards increasingly overparameterized models (c.f., deep learning). In this paper, we study an underexplored hidden cost of overparameterization: the fact that overparameterized models may be more vulnerable to privacy attacks, in particular the membership inference attack that predicts the (potentially sensitive) examples used to train a model. We significantly extend the relatively few empirical results on this problem by theoretically proving for an overparameterized linear regression model in the Gaussian data setting that membership inference vulnerability increases with the number of parameters. Moreover, a range of empirical studies indicates that more complex, nonlinear models exhibit the same behavior.

Recent work has leveraged the popular distributionally robust optimization paradigm to combat overfitting in classical logistic regression. While the resulting classification scheme displays a promising performance in numerical experiments, it is inherently limited to numerical features. In this paper, we show that distributionally robust logistic regression with mixed (\emph{i.e.}, numerical and categorical) features, despite amounting to an optimization problem of exponential size, admits a polynomial-time solution scheme. We subsequently develop a practically efficient cutting plane approach that solves the problem as a sequence of polynomial-time solvable exponential conic programs. Our method retains many of the desirable theoretical features of previous works, but---in contrast to the literature---it does not admit an equivalent representation as a regularized logistic regression, that is, it represents a genuinely novel variant of the logistic regression problem.

We investigate nonlinear instrumental variable (IV) regression given high-dimensional instruments. We propose a simple algorithm which combines kernelized IV methods and an arbitrary, adaptive regression algorithm, accessed as a black box. Our algorithm enjoys faster-rate convergence and adapts to the dimensionality of informative latent features, while avoiding an expensive minimax optimization procedure, which has been necessary to establish similar guarantees. It further brings the benefit of flexible machine learning models to quasi-Bayesian uncertainty quantification, likelihood-based model selection, and model averaging. Simulation studies demonstrate the competitive performance of our method.

Despite the empirical success of deep meta-learning, theoretical understanding of overparameterized meta-learning is still limited. This paper studies the generalization of a widely used meta-learning approach, Model-Agnostic Meta-Learning (MAML), which aims to find a good initialization for fast adaptation to new tasks. Under a mixed linear regression model, we analyze the generalization properties of MAML trained with SGD in the overparameterized regime. We provide both upper and lower bounds for the excess risk of MAML, which captures how SGD dynamics affect these generalization bounds. With such sharp characterizations, we further explore how various learning parameters impact the generalization capability of overparameterized MAML, including explicitly identifying typical data and task distributions that can achieve diminishing generalization error with overparameterization, and characterizing the impact of adaptation learning rate on both excess risk and the early stopping time.

We consider the problem of robustly testing the norm of a high-dimensional sparse signal vector under two different observation models. In the first model, we are given n i.i.d. We show that any algorithm for this task requires n \Omega\left(s\log\frac{ed}{s}\right) samples, which is tight up to logarithmic factors. We also extend our results to other common notions of sparsity, namely, \ \theta\ _q\le s for any 0 q 2 . In the second observation model that we consider, the data is generated according to a sparse linear regression model, where the covariates are i.i.d.

Multivariate regression (or multi-task learning) concerns the task of predicting the value of multiple responses from a set of covariates. In this article, we propose a convex optimization formulation for high-dimensional multivariate linear regression under a general error covariance structure. The main difficulty with simultaneous estimation of the regression coefficients and the error covariance matrix lies in the fact that the negative log-likelihood function is not convex. To overcome this difficulty, a new parameterization is proposed, under which the negative log-likelihood function is proved to be convex. For faster computation, two other alternative loss functions are also considered, and proved to be convex under the proposed parameterization.

Existing neuroimaging methods typically perform either discovery of unknown neural structure or testing of neural structure associated with mental tasks. We therefore propose to blend representation modelling and task classification into a unified statistical learning problem. A multinomial logistic regression is introduced that is constrained by factored coefficients and coupled with an autoencoder. We show that this approach yields more accurate and interpretable neural models of psychological tasks in a reference dataset, as well as better generalization to other datasets.

Linear regression is an important tool across many fields that work with sensitive human-sourced data. Significant prior work has focused on producing differentially private point estimates, which provide a privacy guarantee to individuals while still allowing modelers to draw insights from data by estimating regression coefficients. We investigate the problem of Bayesian linear regression, with the goal of computing posterior distributions that correctly quantify uncertainty given privately released statistics. We show that a naive approach that ignores the noise injected by the privacy mechanism does a poor job in realistic data settings. We then develop noise-aware methods that perform inference over the privacy mechanism and produce correct posteriors across a wide range of scenarios.

The majority of our hypothesis tests are based on differentially private versions of the F -statistic for the general linear model framework, which are uniformly most powerful unbiased in the non-private setting. We also present another test for testing mixtures, based on the differentially private nonparametric tests of Couch, Kazan, Shi, Bray, and Groce (CCS 2019), which is especially suited for the small dataset regime. We show that the differentially private F -statistic converges to the asymptotic distribution of its non-private counterpart. Through a suite of Monte Carlo based experiments, we show that our tests achieve desired \textit{significance levels} and have a high \textit{power} that approaches the power of the non-private tests as we increase sample sizes or the privacy-loss parameter. We also show when our tests outperform existing methods in the literature.

Identifying the relevant variables for a classification model with correct confidence levels is a central but difficult task in high-dimension. Despite the core role of sparse logistic regression in statistics and machine learning, it still lacks a good solution for accurate inference in the regime where the number of features p is as large as or larger than the number of samples n . Here we tackle this problem by improving the Conditional Randomization Test (CRT). The original CRT algorithm shows promise as a way to output p-values while making few assumptions on the distribution of the test statistics. As it comes with a prohibitive computational cost even in mildly high-dimensional problems, faster solutions based on distillation have been proposed.