Does testing remain easier than learning? This type of question, framed in a minimax setting, sits at the intersection of theoretical computer science (where it is captured under the framework of distribution testing) and robust statistics.

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. samples from the distribution $\mathcal{N}\left(\theta,I_d\right)$ (with unknown $\theta$), of which a small fraction has been arbitrarily corrupted. Under the promise that $\|\theta\|_0\le s$, we want to correctly distinguish whether $\|\theta\|_2=0$ or $\|\theta\|_2>\gamma$, for some input parameter $\gamma>0$. 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. Gaussian and the regression coefficient (signal) is known to be $s$-sparse. Here too we assume that an $\epsilon$-fraction of the data is arbitrarily corrupted. We show that any algorithm that reliably tests the norm of the regression coefficient requires at least $n=\Omega\left(\min(s\log d,{1}/{\gamma^4})\right)$ samples. Our results show that the complexity of testing in these two settings significantly increases under robustness constraints. This is in line with the recent observations made in robust mean testing and robust covariance testing.

Classical asymptotic theory for statistical hypothesis testing, for example Wilks' theorem for likelihood ratios, usually involves calibrating the test statistic by fixing the dimension $d$ while letting the sample size $n$ increase to infinity. In the last few decades, a great deal of effort has been dedicated towards understanding how these methods behave in high-dimensional settings, where $d_n$ and $n$ both increase to infinity together at some prescribed relative rate. This often leads to different tests in the two settings, depending on the assumptions about the dimensionality. This leaves the practitioner in a bind: given a dataset with 100 samples in 20 dimensions, should they calibrate by assuming $n \gg d$, or $d_n/n \approx 0.2$? This paper considers the goal of dimension-agnostic inference---developing methods whose validity does not depend on any assumption on $d_n$. We describe one generic approach that uses variational representations of existing test statistics along with sample-splitting and self-normalization (studentization) to produce a Gaussian limiting null distribution. We exemplify this technique for a handful of classical problems, such as one-sample mean testing, testing if a covariance matrix equals the identity, and kernel methods for testing equality of distributions using degenerate U-statistics like the maximum mean discrepancy. Without explicitly targeting the high-dimensional setting, our tests are shown to be minimax rate-optimal, meaning that the power of our tests cannot be improved further up to a constant factor. A hidden advantage is that our proofs are simple and transparent. We end by describing several fruitful open directions.