Breaking the curse of dimensionality in regression

Models with many signals, high-dimensional models, often impose structures on the signal strengths. The common assumption is that only a few signals are strong and most of the signals are zero or close (collectively) to zero. However, such a requirement might not be valid in many real-life applications. In this article, we are interested in conducting large-scale inference in models that might have signals of mixed strengths. The key challenge is that the signals that are not under testing might be collectively non-negligible (although individually small) and cannot be accurately learned. This article develops a new class of tests that arise from a moment matching formulation. A virtue of these moment-matching statistics is their ability to borrow strength across features, adapt to the sparsity size and exert adjustment for testing growing number of hypothesis. GRoup-level Inference of Parameter, GRIP, test harvests effective sparsity structures with hypothesis formulation for an efficient multiple testing procedure. Simulated data showcase that GRIPs error control is far better than the alternative methods. We develop a minimax theory, demonstrating optimality of GRIP for a broad range of models, including those where the model is a mixture of a sparse and high-dimensional dense signals.Comment: 51 page

Post-Selection Inference for Generalized Linear Models with Many Controls

This paper considers generalized linear models in the presence of many controls. We lay out a general methodology to estimate an effect of interest based on the construction of an instrument that immunize against model selection mistakes and apply it to the case of logistic binary choice model. More specifically we propose new methods for estimating and constructing confidence regions for a regression parameter of primary interest $\alpha_0$, a parameter in front of the regressor of interest, such as the treatment variable or a policy variable. These methods allow to estimate $\alpha_0$ at the root-$n$ rate when the total number $p$ of other regressors, called controls, potentially exceed the sample size $n$ using sparsity assumptions. The sparsity assumption means that there is a subset of $s<n$ controls which suffices to accurately approximate the nuisance part of the regression function. Importantly, the estimators and these resulting confidence regions are valid uniformly over $s$-sparse models satisfying $s^2\log^2 p = o(n)$ and other technical conditions. These procedures do not rely on traditional consistent model selection arguments for their validity. In fact, they are robust with respect to moderate model selection mistakes in variable selection. Under suitable conditions, the estimators are semi-parametrically efficient in the sense of attaining the semi-parametric efficiency bounds for the class of models in this paper

Adversarial Training in Affective Computing and Sentiment Analysis: Recent Advances and Perspectives

Over the past few years, adversarial training has become an extremely active research topic and has been successfully applied to various Artificial Intelligence (AI) domains. As a potentially crucial technique for the development of the next generation of emotional AI systems, we herein provide a comprehensive overview of the application of adversarial training to affective computing and sentiment analysis. Various representative adversarial training algorithms are explained and discussed accordingly, aimed at tackling diverse challenges associated with emotional AI systems. Further, we highlight a range of potential future research directions. We expect that this overview will help facilitate the development of adversarial training for affective computing and sentiment analysis in both the academic and industrial communities

Combinatorial Penalties: Which structures are preserved by convex relaxations?

We consider the homogeneous and the non-homogeneous convex relaxations for combinatorial penalty functions defined on support sets. Our study identifies key differences in the tightness of the resulting relaxations through the notion of the lower combinatorial envelope of a set-function along with new necessary conditions for support identification. We then propose a general adaptive estimator for convex monotone regularizers, and derive new sufficient conditions for support recovery in the asymptotic setting

Fast global convergence of gradient methods for high-dimensional statistical recovery

Many statistical $M$-estimators are based on convex optimization problems formed by the combination of a data-dependent loss function with a norm-based regularizer. We analyze the convergence rates of projected gradient and composite gradient methods for solving such problems, working within a high-dimensional framework that allows the data dimension \pdim to grow with (and possibly exceed) the sample size \numobs. This high-dimensional structure precludes the usual global assumptions---namely, strong convexity and smoothness conditions---that underlie much of classical optimization analysis. We define appropriately restricted versions of these conditions, and show that they are satisfied with high probability for various statistical models. Under these conditions, our theory guarantees that projected gradient descent has a globally geometric rate of convergence up to the \emph{statistical precision} of the model, meaning the typical distance between the true unknown parameter $\theta^*$ and an optimal solution $\hat{\theta}$. This result is substantially sharper than previous convergence results, which yielded sublinear convergence, or linear convergence only up to the noise level. Our analysis applies to a wide range of $M$-estimators and statistical models, including sparse linear regression using Lasso ($\ell_1$-regularized regression); group Lasso for block sparsity; log-linear models with regularization; low-rank matrix recovery using nuclear norm regularization; and matrix decomposition. Overall, our analysis reveals interesting connections between statistical precision and computational efficiency in high-dimensional estimation

Estimating individual treatment effects under unobserved confounding using binary instruments

Estimating individual treatment effects (ITEs) from observational data is relevant in many fields such as personalized medicine. However, in practice, the treatment assignment is usually confounded by unobserved variables and thus introduces bias. A remedy to remove the bias is the use of instrumental variables (IVs). Such settings are widespread in medicine (e.g., trials where compliance is used as binary IV). In this paper, we propose a novel, multiply robust machine learning framework, called MRIV, for estimating ITEs using binary IVs and thus yield an unbiased ITE estimator. Different from previous work for binary IVs, our framework estimates the ITE directly via a pseudo outcome regression. (1) We provide a theoretical analysis where we show that our framework yields multiply robust convergence rates: our ITE estimator achieves fast convergence even if several nuisance estimators converge slowly. (2) We further show that our framework asymptotically outperforms state-of-the-art plug-in IV methods for ITE estimation. (3) We build upon our theoretical results and propose a tailored deep neural network architecture called MRIV-Net for ITE estimation using binary IVs. Across various computational experiments, we demonstrate empirically that our MRIV-Net achieves state-of-the-art performance. To the best of our knowledge, our MRIV is the first machine learning framework for estimating ITEs in the binary IV setting shown to be multiply robust

High-dimensional semi-supervised learning: in search for optimal inference of the mean

We provide a high-dimensional semi-supervised inference framework focused on the mean and variance of the response. Our data are comprised of an extensive set of observations regarding the covariate vectors and a much smaller set of labeled observations where we observe both the response as well as the covariates. We allow the size of the covariates to be much larger than the sample size and impose weak conditions on a statistical form of the data. We provide new estimators of the mean and variance of the response that extend some of the recent results presented in low-dimensional models. In particular, at times we will not necessitate consistent estimation of the functional form of the data. Together with estimation of the population mean and variance, we provide their asymptotic distribution and confidence intervals where we showcase gains in efficiency compared to the sample mean and variance. Our procedure, with minor modifications, is then presented to make important contributions regarding inference about average treatment effects. We also investigate the robustness of estimation and coverage and showcase widespread applicability and generality of the proposed method