We explore the connection between outlier-robust high-dimensional statistics and non-convex optimization in the presence of sparsity constraints, with a focus on the fundamental tasks of robust sparse mean estimation and robust sparse PCA. We develop novel and simple optimization formulations for these problems such that any approximate stationary point of the associated optimization problem yields a near-optimal solution for the underlying robust estimation task. As a corollary, we obtain that any first-order method that efficiently converges to stationarity yields an efficient algorithm for these tasks.1 The obtained algorithms are simple, practical, and succeed under broader distributional assumptions compared to prior work.

Estimation and inference on causal parameters is typically reduced to a generalized method of moments problem, which involves auxiliary functions that correspond to solutions to a regression or classification problem. Recent line of work on debiased machine learning shows how one can use generic machine learning estimators for these auxiliary problems, while maintaining asymptotic normality and root-n consistency of the target parameter of interest, while only requiring mean-squared-error guarantees from the auxiliary estimation algorithms. The literature typically requires that these auxiliary problems are fitted on a separate sample or in a cross-fitting manner. We show that when these auxiliary estimation algorithms satisfy natural leave-one-out stability properties, then sample splitting is not required. This allows for sample re-use, which can be beneficial in moderately sized sample regimes. For instance, we show that the stability properties that we propose are satisfied for ensemble bagged estimators, built via sub-sampling without replacement, a popular technique in machine learning practice.

Sequential data collection has emerged as a widely adopted technique for enhancing the efficiency of data gathering processes. Despite its advantages, such data collection mechanism often introduces complexities to the statistical inference procedure.

Each block represents atime-invariant iterativeprocess as the first layer in thei-th block,xi(1), is unrolled into a pattern-dependent number,Ki, of processing stages, using weight matricesAi andBi. The skip connections from the input,ui, to all layers in blockimake the process nonautonomous. Blocks can be chained together (each block modeling adifferent latent space) by passing final latentrepresentation,xi(Ki),ofblockiastheinputtoblocki+1.

This progress has led to a revival of robust statistics from an algorithmic perspective (see, e.g., [DK19, DKK+21] for surveys on the topic).

Statistical inference in contextual bandits is complicated by the adaptive, non-i.i.d. nature of the data. A growing body of work has shown that classical least-squares inference may fail under adaptive sampling, and that constructing valid confidence intervals for linear functionals of the model parameter typically requires paying an unavoidable inflation of order $\sqrt{d \log T}$. This phenomenon -- often referred to as the price of adaptivity -- highlights the inherent difficulty of reliable inference under general contextual bandit policies. A key structural property that circumvents this limitation is the \emph{stability} condition of Lai and Wei, which requires the empirical feature covariance to concentrate around a deterministic limit. When stability holds, the ordinary least-squares estimator satisfies a central limit theorem, and classical Wald-type confidence intervals -- designed for i.i.d. data -- become asymptotically valid even under adaptation, \emph{without} incurring the $\sqrt{d \log T}$ price of adaptivity. In this paper, we propose and analyze a penalized EXP4 algorithm for linear contextual bandits. Our first main result shows that this procedure satisfies the Lai--Wei stability condition and therefore admits valid Wald-type confidence intervals for linear functionals. Our second result establishes that the same algorithm achieves regret guarantees that are minimax optimal up to logarithmic factors, demonstrating that stability and statistical efficiency can coexist within a single contextual bandit method. Finally, we complement our theory with simulations illustrating the empirical normality of the resulting estimators and the sharpness of the corresponding confidence intervals.

Recently, several researchers have started to view VDNNs from a dynamical systems perspective.