We study the statistical-computational tradeoffs in a high dimensional single index modelY = f(X>β)+, where f is unknown,X is a Gaussian vector and β is s-sparse with unit norm. WhenCov(Y,X>β) 6= 0, [43] shows that the direction and support ofβ can be recovered using a generalized version of Lasso.

For the first question, we introduce the notions of first-and second-order Stein's associations which

We revisit the discrete argmin inference problem in high-dimensional settings. Given $n$ observations from a $d$ dimensional vector, the goal is to test whether the $r$th component of the mean vector is the smallest among all components. We propose dimension-agnostic tests that maintain validity regardless of how $d$ scales with $n$, and regardless of arbitrary ties in the mean vector. Notably, our validity holds under mild moment conditions, requiring little more than finiteness of a second moment, and permitting possibly strong dependence between coordinates. In addition, we establish the local minimax separation rate for this problem, which adapts to the cardinality of a confusion set, and show that the proposed tests attain this rate. Our method uses the sample splitting and self-normalization approach of Kim and Ramdas (2024). Our tests can be easily inverted to yield confidence sets for the argmin index. Empirical results illustrate the strong performance of our approach in terms of type I error control and power compared to existing methods.

Federated learning has attracted significant recent attention due to its applicability across a wide range of settings where data is collected and analyzed across disparate locations. In this paper, we study federated nonparametric goodness-of-fit testing in the white-noise-with-drift model under distributed differential privacy (DP) constraints. We first establish matching lower and upper bounds, up to a logarithmic factor, on the minimax separation rate. This optimal rate serves as a benchmark for the difficulty of the testing problem, factoring in model characteristics such as the number of observations, noise level, and regularity of the signal class, along with the strictness of the $(\epsilon,\delta)$-DP requirement. The results demonstrate interesting and novel phase transition phenomena. Furthermore, the results reveal an interesting phenomenon that distributed one-shot protocols with access to shared randomness outperform those without access to shared randomness. We also construct a data-driven testing procedure that possesses the ability to adapt to an unknown regularity parameter over a large collection of function classes with minimal additional cost, all while maintaining adherence to the same set of DP constraints.

In the interest of interpretability, computation, and circumventing the statistical curse of dimensionality plaguing high dimensional regression, structure is often assumed on the true regression function. Indeed, it might plausibly be argued that sparse linear regression is the distinguishing export of modern statistics. Despite its popularity, circumstances may call for more flexibility to capture nonlinear effects of the covariates. Striking a balance between flexibility and structure, Hastie and Tibshirani [19] proposed generalized additive models (GAMs) as a natural extension to the vaunted linear model. In a GAM, the regression function admits an additive decomposition of univariate (nonlinear) component functions. However, as in the linear model, the sample size must outpace the dimension for consistent estimation. Following modern statistical instinct, a sparse additive model is compelling [28, 32, 34, 37, 38, 47]. The regression function admits an additive decomposition of univariate functions for which only a small subset are nonzero; it is the combination of a GAM and sparsity.

Finding anonymization mechanisms to protect personal data is at the heart of machine learning research. Here we consider the consequences of local differential privacy constraints on goodness-of-fit testing, i.e. the statistical problem assessing whether sample points are generated from a fixed density $f_0$, or not. The observations are hidden and replaced by a stochastic transformation satisfying the local differential privacy constraint. In this setting, we propose a new testing procedure which is based on an estimation of the quadratic distance between the density $f$ of the unobserved sample and $f_0$. We establish minimax separation rates for our test over Besov balls. We also provide a lower bound, proving the optimality of our result. To the best of our knowledge, we provide the first minimax optimal test and associated private transformation under a local differential privacy constraint, quantifying the price to pay for data privacy.

We study the problem of independence testing given independent and identically distributed pairs taking values in a $\sigma$-finite, separable measure space. Defining a natural measure of dependence $D(f)$ as the squared $L_2$-distance between a joint density $f$ and the product of its marginals, we first show that there is no valid test of independence that is uniformly consistent against alternatives of the form $\{f: D(f) \geq \rho^2 \}$. We therefore restrict attention to alternatives that impose additional Sobolev-type smoothness constraints, and define a permutation test based on a basis expansion and a $U$-statistic estimator of $D(f)$ that we prove is minimax optimal in terms of its separation rates in many instances. Finally, for the case of a Fourier basis on $[0,1]^2$, we provide an approximation to the power function that offers several additional insights.