influence function analysis

Constructing efficient debiased estimators traditionally requires manual derivation of the efficient influence function (EIF), a labor-intensive process with high technical barriers and poor scalability. Method: This paper introduces Dimple, the first framework that models statistical functionals as compositions of differentiable primitives satisfying a novel differentiability condition; it leverages automatic differentiation to directly generate unbiased, efficient estimators while simultaneously identifying nuisance parameters. Dimple integrates probabilistic programming with functional decomposition, eliminating the need for explicit EIF derivation. Contribution/Results: We provide an open-source Python library enabling users to define parameters, generate estimators, and perform inference in just a few lines of code. Extensive experiments demonstrate Dimple’s effectiveness across diverse causal and semiparametric models—including AIPW, DR-Learner, and doubly robust IV—significantly lowering the barrier to constructing efficient estimators without sacrificing statistical efficiency.

This work investigates the impact of infinitesimal perturbations to training data on model performance, aiming to enhance model interpretability and robustness. Addressing the high computational cost and limited interpretability of conventional influence functions—particularly in non-convex models—the paper pioneers the integration of Fisher information geometry into influence estimation. It introduces the Approximate Fisher Influence Function (AFIF), reformulating influence estimation as a weighted empirical risk minimization problem. Leveraging information-geometric principles, the authors derive an efficient approximation algorithm that avoids explicit Hessian inversion. The method achieves several-fold speedup over Newton-type approaches while maintaining high accuracy and strong robustness across both generalized linear models and non-convex neural networks. By unifying geometric insight with practical scalability, AFIF bridges interpretability and usability in influence analysis.

In causal mediation analysis, conventional estimators of the mediated effect functional suffer from low accuracy and high sensitivity to misspecification of nuisance functions. To address this, we propose a bias-structure-guided two-stage framework that decouples nuisance function estimation. In Stage I, we estimate only the bias-relevant component of the mediation mechanism—rather than the full mechanism—thereby reducing model dependence. In Stage II, we introduce a nonparametric weighted balancing estimator, where weights are constructed by directly optimizing the asymptotic bias of the mediated effect estimator. We establish theoretical guarantees: the resulting estimator is consistent and asymptotically normal, and remains robust under partial misspecification of nuisance functions. Compared with standard approaches, our method substantially improves estimation accuracy and reliability. It provides a principled tool for mediation inference in high-dimensional settings or under model uncertainty.

This work addresses the robustness of mean estimation in statistical learning under three concurrent challenges: adversarial data contamination, heavy-tailed distributions, and differential privacy constraints. Methodologically, it unifies robust statistics, high-dimensional geometry, stochastic optimization, and differential privacy theory to establish the first conceptual and algorithmic bridge across distinct robustness paradigms. Key technical abstractions—including iterative filtering, covariance trimming, and fractional gradient descent—are identified as common algorithmic primitives. The paper proposes a suite of computationally efficient estimators achieving statistically optimal convergence rates; each attains the information-theoretic lower bound under all three constraint classes simultaneously. By reconciling theoretical tightness with practical efficiency, this framework advances robust mean estimation from ad hoc heuristics toward a principled, unified design paradigm.

This paper addresses statistical inference for smooth functionals of nonparametric M-estimators—such as causal effects, quantiles, and survival functions—by proposing the autoDML framework, which automates debiasing without manual influence function derivation. Methodologically, it introduces the first fully automated influence function construction mechanism, integrating gradient/Hessian estimation of the loss, Riesz representer learning, joint risk minimization, and targeted minimum loss estimation; it supports vector-valued M-estimators and Neyman-orthogonal losses. Theoretically, autoDML ensures double robustness and robustness to model misspecification, achieving semiparametric efficiency and second-order bias suppression under quadratic risk. Empirically, it is validated on long-term survival probability estimation in a semiparametric beta-geometric model, demonstrating substantial improvements in both inferential accuracy and automation.

This work addresses the lack of intuitive geometric interpretation in classical semiparametric efficiency theory, which has hindered the derivation and understanding of influence functions. The paper reformulates the theory within a differential geometric framework on the space of probability distributions, drawing an analogy to multivariate calculus: statistical paths, scores, and influence functions correspond respectively to curves, velocity vectors, and gradients. It demonstrates that the efficient influence function arises naturally as an orthogonal projection. By integrating functional analysis, differential geometry, and statistical inference, the study establishes a unified geometric interpretation of scores, tangent spaces, nuisance tangent spaces, and efficient influence functions. This synthesis not only clarifies several foundational theoretical issues but also substantially enhances the interpretability of methods in causal inference and missing data analysis.

This work addresses the lack of precise characterization of individual sample influence in high-dimensional convex M-estimation when the sample size and dimensionality are of the same order (\(n \sim d\)). Under Gaussian design assumptions, the authors rigorously derive the asymptotic distribution of influence measures by integrating high-dimensional asymptotic analysis, random matrix theory, and leave-one-out influence diagnostics. They establish that the influence measure converges to an explicitly expressible limiting distribution. The analysis further reveals that highly influential samples concentrate near the decision boundary, thereby providing theoretical justification for boundary-based heuristic strategies in active learning and uncovering an intrinsic connection between influence analysis and classification boundaries.

This work addresses the infeasibility of computing influence functions when model size vastly exceeds dataset size. The authors propose a dual representation–based approach to influence function estimation, incorporating kernel methods into influence analysis. Under the assumption of linearizable models, they explicitly construct a dual formulation that reduces computational complexity to scale with the dataset size rather than the number of model parameters. This framework represents the first efficient influence computation method whose complexity is governed by data size, enabling accurate estimation of the effect of removing individual data points on model parameters, predictions, and loss—particularly advantageous in large-model, small-data regimes where conventional approaches incur prohibitive computational costs.

This study addresses statistical inference challenges arising from weak identification, noise contamination, multiple constraints, and model misspecification by proposing a unified framework based on Lagrangian constrained optimization. The work innovatively introduces Individual Shadow Prices (ISPs) to quantify the information content of each constraint and designs platform rules to distinguish signal from noise. Incorporating a Stein-type risk criterion, the method employs a data-driven approach to select tolerance parameters and leverages Karush–Kuhn–Tucker (KKT) conditions to achieve debiased estimation. Theoretical analysis establishes the consistency and asymptotic normality of the resulting estimator. Numerical simulations and empirical application to the Solow growth model demonstrate that the proposed approach effectively captures model uncertainty and enhances inference accuracy.