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1. Interval Estimation of Individual-Level Causal Effects Under Unobserved Confounding

   Kallus, Nathan; Mao, Xiaojie; Zhou, Angela (January 2019, Proceedings of Machine Learning Research)

   We study the problem of learning conditional average treatment effects (CATE) from observational data with unobserved confounders. The CATE function maps baseline covariates to individual causal effect predictions and is key for personalized assessments. Recent work has focused on how to learn CATE under unconfoundedness, i.e., when there are no unobserved confounders. Since CATE may not be identified when unconfoundedness is violated, we develop a functional interval estimator that predicts bounds on the individual causal effects under realistic violations of unconfoundedness. Our estimator takes the form of a weighted kernel estimator with weights that vary adversarially. We prove that our estimator is sharp in that it converges exactly to the tightest bounds possible on CATE when there may be unobserved confounders. Further, we study personalized decision rules derived from our estimator and prove that they achieve optimal minimax regret asymptotically. We assess our approach in a simulation study as well as demonstrate its application in the case of hormone replacement therapy by comparing conclusions from a real observational study and clinical trial. 

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2. Nearly Optimal Learning Using Sparse Deep ReLU Networks in Regularized Empirical Risk Minimization With Lipschitz Loss

   https://doi.org/10.1162/neco_a_01742  

   Huang, Ke; Liu, Mingming; Ma, Shujie (March 2025, Neural Computation)

   Abstract We propose a sparse deep ReLU network (SDRN) estimator of the regression function obtained from regularized empirical risk minimization with a Lipschitz loss function. Our framework can be applied to a variety of regression and classification problems. We establish novel nonasymptotic excess risk bounds for our SDRN estimator when the regression function belongs to a Sobolev space with mixed derivatives. We obtain a new, nearly optimal, risk rate in the sense that the SDRN estimator can achieve nearly the same optimal minimax convergence rate as one-dimensional nonparametric regression with the dimension involved in a logarithm term only when the feature dimension is fixed. The estimator has a slightly slower rate when the dimension grows with the sample size. We show that the depth of the SDRN estimator grows with the sample size in logarithmic order, and the total number of nodes and weights grows in polynomial order of the sample size to have the nearly optimal risk rate. The proposed SDRN can go deeper with fewer parameters to well estimate the regression and overcome the overfitting problem encountered by conventional feedforward neural networks. 

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3. Optimal Bayesian estimation for random dot product graphs

   https://doi.org/10.1093/biomet/asaa031  

   Xie, Fangzheng; Xu, Yanxun (July 2020, Biometrika)

   Summary We propose and prove the optimality of a Bayesian approach for estimating the latent positions in random dot product graphs, which we call posterior spectral embedding. Unlike classical spectral-based adjacency, or Laplacian spectral embedding, posterior spectral embedding is a fully likelihood-based graph estimation method that takes advantage of the Bernoulli likelihood information of the observed adjacency matrix. We develop a minimax lower bound for estimating the latent positions, and show that posterior spectral embedding achieves this lower bound in the following two senses: it both results in a minimax-optimal posterior contraction rate and yields a point estimator achieving the minimax risk asymptotically. The convergence results are subsequently applied to clustering in stochastic block models with positive semidefinite block probability matrices, strengthening an existing result concerning the number of misclustered vertices. We also study a spectral-based Gaussian spectral embedding as a natural Bayesian analogue of adjacency spectral embedding, but the resulting posterior contraction rate is suboptimal by an extra logarithmic factor. The practical performance of the proposed methodology is illustrated through extensive synthetic examples and the analysis of Wikipedia graph data. 

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4. Nonparametric, Tuning-Free Estimation of S-Shaped Functions

   https://doi.org/10.1111/rssb.12481  

   Feng, Oliver Y.; Chen, Yining; Han, Qiyang; Carroll, Raymond J.; Samworth, Richard J. (April 2022, Journal of the Royal Statistical Society Series B: Statistical Methodology)

   Abstract We consider the nonparametric estimation of an S-shaped regression function. The least squares estimator provides a very natural, tuning-free approach, but results in a non-convex optimization problem, since the inflection point is unknown. We show that the estimator may nevertheless be regarded as a projection onto a finite union of convex cones, which allows us to propose a mixed primal-dual bases algorithm for its efficient, sequential computation. After developing a projection framework that demonstrates the consistency and robustness to misspecification of the estimator, our main theoretical results provide sharp oracle inequalities that yield worst-case and adaptive risk bounds for the estimation of the regression function, as well as a rate of convergence for the estimation of the inflection point. These results reveal not only that the estimator achieves the minimax optimal rate of convergence for both the estimation of the regression function and its inflection point (up to a logarithmic factor in the latter case), but also that it is able to achieve an almost-parametric rate when the true regression function is piecewise affine with not too many affine pieces. Simulations and a real data application to air pollution modelling also confirm the desirable finite-sample properties of the estimator, and our algorithm is implemented in the R package Sshaped. 

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5. Adaptive minimax density estimation on ℝ d for Huber’s contamination model

   https://doi.org/10.1093/imaiai/iaad045  

   Zhang, Peiliang; Ren, Zhao (November 2023, Information and Inference: A Journal of the IMA)

   Abstract We address the problem of adaptive minimax density estimation on $$\mathbb{R}^{d}$$ with $$L_{p}$$ loss functions under Huber’s contamination model. To investigate the contamination effect on the optimal estimation of the density, we first establish the minimax rate with the assumption that the density is in an anisotropic Nikol’skii class. We then develop a data-driven bandwidth selection procedure for kernel estimators, which can be viewed as a robust generalization of the Goldenshluger-Lepski method. We show that the proposed bandwidth selection rule can lead to the estimator being minimax adaptive to either the smoothness parameter or the contamination proportion. When both of them are unknown, we prove that finding any minimax-rate adaptive method is impossible. Extensions to smooth contamination cases are also discussed. 

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