The increasing complexity of machine learning (ML) and artificial intelligence (AI) models has created a pressing need for tools that help scientists, engineers, and policymakers interpret and refine model decisions and predictions. Influence functions, originating from robust statistics, have emerged as a popular approach for this purpose. However, the heuristic foundations of influence functions rely on low-dimensional assumptions where the number of parameters $p$ is much smaller than the number of observations $n$. In contrast, modern AI models often operate in high-dimensional regimes with large $p$, challenging these assumptions. In this paper, we examine the accuracy of influence functions in high-dimensional settings. Our theoretical and empirical analyses reveal that influence functions cannot reliably fulfill their intended purpose. We then introduce an alternative approximation, called Newfluence, that maintains similar computational efficiency while offering significantly improved accuracy. Newfluence is expected to provide more accurate insights than many existing methods for interpreting complex AI models and diagnosing their issues. Moreover, the high-dimensional framework we develop in this paper can also be applied to analyze other popular techniques, such as Shapley values.

Machine unlearning focuses on the computationally efficient removal of specific training data from trained models, ensuring that the influence of forgotten data is effectively eliminated without the need for full retraining. Despite advances in low-dimensional settings, where the number of parameters \( p \) is much smaller than the sample size \( n \), extending similar theoretical guarantees to high-dimensional regimes remains challenging. We propose an unlearning algorithm that starts from the original model parameters and performs a theory-guided sequence of Newton steps \( T \in \{ 1,2\}\). After this update, carefully scaled isotropic Laplacian noise is added to the estimate to ensure that any (potential) residual influence of forget data is completely removed. We show that when both \( n, p \to \infty \) with a fixed ratio \( n/p \), significant theoretical and computational obstacles arise due to the interplay between the complexity of the model and the finite signal-to-noise ratio. Finally, we show that, unlike in low-dimensional settings, a single Newton step is insufficient for effective unlearning in high-dimensional problems -- however, two steps are enough to achieve the desired certifiebility. We provide numerical experiments to support the certifiability and accuracy claims of this approach.

Despite a large and significant body of recent work focused on estimating the out-of-sample risk of regularized models in the high dimensional regime, a theoretical understanding of this problem for non-differentiable penalties such as generalized LASSO and nuclear norm is missing. In this paper we resolve this challenge. We study this problem in the proportional high dimensional regime where both the sample size n and number of features p are large, and n/p and the signal-to-noise ratio (per observation) remain finite. We provide finite sample upper bounds on the expected squared error of leave-one-out cross-validation (LO) in estimating the out-of-sample risk. The theoretical framework presented here provides a solid foundation for elucidating empirical findings that show the accuracy of LO.

The out-of-sample error (OO) is the main quantity of interest in risk estimation and model selection. Leave-one-out cross validation (LO) offers a (nearly) distribution-free yet computationally demanding approach to estimate OO. Recent theoretical work showed that approximate leave-one-out cross validation (ALO) is a computationally efficient and statistically reliable estimate of LO (and OO) for generalized linear models with differentiable regularizers. For problems involving non-differentiable regularizers, despite significant empirical evidence, the theoretical understanding of ALO's error remains unknown. In this paper, we present a novel theory for a wide class of problems in the generalized linear model family with non-differentiable regularizers. We bound the error |ALO - LO| in terms of intuitive metrics such as the size of leave-i-out perturbations in active sets, sample size n, number of features p and regularization parameters. As a consequence, for the $\ell_1$-regularized problems, we show that |ALO - LO| goes to zero as p goes to infinity while n/p and SNR are fixed and bounded.

We study the problem of out-of-sample risk estimation in the high dimensional regime where both the sample size $n$ and number of features $p$ are large, and $n/p$ can be less than one. Extensive empirical evidence confirms the accuracy of leave-one-out cross validation (LO) for out-of-sample risk estimation. Yet, a unifying theoretical evaluation of the accuracy of LO in high-dimensional problems has remained an open problem. This paper aims to fill this gap for penalized regression in the generalized linear family. With minor assumptions about the data generating process, and without any sparsity assumptions on the regression coefficients, our theoretical analysis obtains finite sample upper bounds on the expected squared error of LO in estimating the out-of-sample error. Our bounds show that the error goes to zero as $n,p \rightarrow \infty$, even when the dimension $p$ of the feature vectors is comparable with or greater than the sample size $n$. One technical advantage of the theory is that it can be used to clarify and connect some results from the recent literature on scalable approximate LO.