Directed Cyclic Graph for Causal Discovery from Multivariate Functional Data

Discovering causal relationship using multivariate functional data has received a significant amount of attention very recently. In this article, we introduce a functional linear structural equation model for causal structure learning when the underlying graph involving the multivariate functions may have cycles. To enhance interpretability, our model involves a low-dimensional causal embedded space such that all the relevant causal information in the multivariate functional data is preserved in this lower-dimensional subspace. We prove that the proposed model is causally identifiable under standard assumptions that are often made in the causal discovery literature. To carry out inference of our model, we develop a fully Bayesian framework with suitable prior specifications and uncertainty quantification through posterior summaries. We illustrate the superior performance of our method over existing methods in terms of causal graph estimation through extensive simulation studies. We also demonstrate the proposed method using a brain EEG dataset.

On the Computational Complexity of Private High-dimensional Model Selection via the Exponential Mechanism

We consider the problem of model selection in a high-dimensional sparse linear regression model under the differential privacy framework. In particular, we consider the problem of differentially private best subset selection and study its utility guarantee. We adopt the well-known exponential mechanism for selecting the best model, and under a certain margin condition, we establish its strong model recovery property. However, the exponential search space of the exponential mechanism poses a serious computational bottleneck. To overcome this challenge, we propose a Metropolis-Hastings algorithm for the sampling step and establish its polynomial mixing time to its stationary distribution in the problem parameters $n,p$, and $s$. Furthermore, we also establish approximate differential privacy for the final estimates of the Metropolis-Hastings random walk using its mixing property. Finally, we also perform some illustrative simulations that echo the theoretical findings of our main results.

Tale of two c(omplex)ities

For decades, best subset selection (BSS) has eluded statisticians mainly due to its computational bottleneck. However, until recently, modern computational breakthroughs have rekindled theoretical interest in BSS and have led to new findings. Recently, Guo et al. (2020) showed that the model selection performance of BSS is governed by a margin quantity that is robust to the design dependence, unlike modern methods such as LASSO, SCAD, MCP, etc. Motivated by their theoretical results, in this paper, we also study the variable selection properties of best subset selection for high-dimensional sparse linear regression setup. We show that apart from the identifiability margin, the following two complexity measures play a fundamental role in characterizing the margin condition for model consistency: (a) complexity of residualized features, (b) complexity of spurious projections. In particular, we establish a simple margin condition that only depends only on the identifiability margin quantity and the dominating one of the two complexity measures. Furthermore, we show that a similar margin condition depending on similar margin quantity and complexity measures is also necessary for model consistency of BSS. For a broader understanding of the complexity measures, we also consider some simple illustrative examples to demonstrate the variation in the complexity measures which broadens our theoretical understanding of the model selection performance of BSS under different correlation structures.

On Consistency and Asymptotic Normality of Least Absolute Deviation Estimators for 2-dimensional Sinusoidal Model

Estimation of the parameters of a 2-dimensional sinusoidal model is a fundamental problem in digital signal processing. In this paper, we propose a robust least absolute deviation (LAD) estimators for parameter estimation. The proposed methodology provides a robust alternative to non-robust estimation techniques like the least squares estimators, in situations where outliers are present in the data or in the presence of heavy tailed noise. We study important asymptotic properties of the LAD estimators and establish the strong consistency and asymptotic normality of the LAD estimators. We further illustrate the advantage of using LAD estimators over least squares estimators through extensive simulation studies.

Thompson Sampling for High-Dimensional Sparse Linear Contextual Bandits

We consider the stochastic linear contextual bandit problem with high-dimensional features. We analyze the Thompson sampling (TS) algorithm, using special classes of sparsity-inducing priors (e.g. spike-and-slab) to model the unknown parameter, and provide a nearly optimal upper bound on the expected cumulative regret. To the best of our knowledge, this is the first work that provides theoretical guarantees of Thompson sampling in high dimensional and sparse contextual bandits. For faster computation, we use spike-and-slab prior to model the unknown parameter and variational inference instead of MCMC to approximate the posterior distribution. Extensive simulations demonstrate improved performance of our proposed algorithm over existing ones.

How does overparametrization affect performance on minority groups?

The benefits of overparameterization for the overall performance of modern machine learning (ML) models are well known. However, the effect of overparameterization at a more granular level of data subgroups is less understood. Recent empirical studies demonstrate encouraging results: (i) when groups are not known, overparameterized models trained with empirical risk minimization (ERM) perform better on minority groups; (ii) when groups are known, ERM on data subsampled to equalize group sizes yields state-of-the-art worst-group-accuracy in the overparameterized regime. In this paper, we complement these empirical studies with a theoretical investigation of the risk of overparameterized random feature models on minority groups. In a setting in which the regression functions for the majority and minority groups are different, we show that overparameterization always improves minority group performance.