We study the problem of learning influence functions under incomplete observations of node activations. Incomplete observations are a major concern as most (online and real-world) social networks are not fully observable. We establish both proper and improper PAC learnability of influence functions under randomly missing observations. Proper PAC learnability under the Discrete-Time Linear Threshold (DLT) and Discrete-Time Independent Cascade (DIC) models is established by reducing incomplete observations to complete observations in a modified graph. Our improper PAC learnability result applies for the DLT and DIC models as well as the Continuous-Time Independent Cascade (CIC) model. It is based on a parametrization in terms of reachability features, and also gives rise to an efficient and practical heuristic. Experiments on synthetic and real-world datasets demonstrate the ability of our method to compensate even for a fairly large fraction of missing observations.

Neural Information Processing Systems http://nips.cc/

Influence functions estimate the effect of removing a training point on a model without theneedtoretrain. Theyarebasedonafirst-order Taylorapproximation thatisguaranteed tobeaccurate forsufficiently small changes tothemodel, and so are commonly used to study the effect of individual points in large datasets. However, we often want to study the effects of largegroups of training points, e.g., todiagnose batch effects orapportion credit between different data sources.

As the amount of data increases, the question arises as to how best to deal with the large datasets. While computational platforms such as Spark [28] and Ray [23] help process large datasets once a desired model is chosen, simply using smaller data can be a faster solution for exploratory data modeling, rapid prototyping, or other tasks where the accuracy obtainable from the full dataset is notneeded.

We propose nonparametric identification and semiparametric estimation of joint potential outcome distributions in the presence of confounding. First, in settings with observed confounding, we derive tighter, covariate-informed bounds on the joint distribution by leveraging conditional copulas. To overcome the non-differentiability of bounding min/max operators, we establish the asymptotic properties for both a direct estimator with polynomial margin condition and a smooth approximation with log-sum-exp operator, facilitating valid inference for individual-level effects under the canonical rank-preserving assumption. Second, we tackle the challenge of unmeasured confounding by introducing a causal representation learning framework. By utilizing instrumental variables, we prove the nonparametric identifiability of the latent confounding subspace under injectivity and completeness conditions. We develop a ``triple machine learning" estimator that employs cross-fitting scheme to sequentially handle the learned representation, nuisance parameters, and target functional. We characterize the asymptotic distribution with variance inflation induced by representation learning error, and provide conditions for semiparametric efficiency. We also propose a practical VAE-based algorithm for confounding representation learning. Simulations and real-world analysis validate the effectiveness of proposed methods. By bridging classical semiparametric theory with modern representation learning, this work provides a robust statistical foundation for distributional and counterfactual inference in complex causal systems.

Traditional IM approaches consider each user/node independently as a potential target customer.

Subset Selection (MISS) problem, which aims to identify a subset of training samples with the greatest collective influence.