Estimators

We study the problem of selecting the best heterogeneous treatment effect (HTE) estimator from a collection of candidates in settings where the treatment effect is fundamentally unobserved. We cast estimator selection as a multiple testing problem and introduce a ground-truth-free procedure based on a cross-fitted, exponentially weighted test statistic. A key component of our method is a two-way sample splitting scheme that decouples nuisance estimation from weight learning and ensures the stability required for valid inference. Leveraging a stability-based central limit theorem, we establish asymptotic familywise error rate control under mild regularity conditions. Empirically, our procedure provides reliable error control while substantially reducing false selections compared with commonly used methods across ACIC 2016, IHDP, and Twins benchmarks, demonstrating that our method is feasible and powerful even without ground-truth treatment effects.

Regression with non-Euclidean responses---e.g., probability distributions, networks, symmetric positive-definite matrices, and compositions---has become increasingly important in modern applications. In this paper, we propose deep Fréchet neural networks (DFNNs), an end-to-end deep learning framework for predicting non-Euclidean responses---which are considered as random objects in a metric space---from Euclidean predictors. Our method utilizes the representation-learning power of deep neural networks (DNNs) to the task of approximating conditional Fréchet means of the response given the predictors, the metric-space analogue of conditional expectations, by minimizing a Fréchet risk. The framework is highly flexible, accommodating diverse metrics and high-dimensional predictors. We establish a universal approximation theorem for DFNNs, advancing the state-of-the-art of neural network approximation theory to general metric-space-valued responses, without making model assumptions or relying on local smoothing. We further establish rigorous generalization guarantees for DFNNs and derive corresponding risk bounds, providing, to the best of our knowledge, the first such theoretical results for deep learning regression with metric-space-valued responses. Empirical studies on synthetic distributional and network-valued responses, as well as real-world applications to predicting compositional responses in an Aitchison simplex and spherical responses, demonstrate that DFNNs consistently outperform all existing methods.

Text watermarks in large language models (LLMs) are an increasingly important tool for detecting synthetic text and distinguishing human-written content from LLM-generated text. While most existing studies focus on determining whether entire texts are watermarked, many real-world scenarios involve mixed-source texts, which blend human-written and watermarked content. In this paper, we address the problem of optimally estimating the watermark proportion in mixed-source texts. We cast this problem as estimating the proportion parameter in a mixture model based on \emph{pivotal statistics}. First, we show that this parameter is not even identifiable in certain watermarking schemes, let alone consistently estimable. In stark contrast, for watermarking methods that employ continuous pivotal statistics for detection, we demonstrate that the proportion parameter is identifiable under mild conditions. We propose efficient estimators for this class of methods, which include several popular unbiased watermarks as examples, and derive minimax lower bounds for any measurable estimator based on pivotal statistics, showing that our estimators achieve these lower bounds. Through evaluations on both synthetic data and mixed-source text generated by open-source models, we demonstrate that our proposed estimators consistently achieve high estimation accuracy.

Prediction-Powered Inference (PPI) is a popular strategy for combining gold-standard and possibly noisy pseudo-labels to perform statistical estimation. Prior work has shown an asymptotic \enquote{free lunch} for PPI++, an adaptive form of PPI, showing that the \textit{asymptotic} variance of PPI++ is always less than or equal to the variance obtained from using gold-standard labels alone. Notably, this result holds \textit{regardless of the quality of the pseudo-labels}. In this work, we demystify this result by conducting an exact finite-sample analysis of the estimation error of PPI++ on the mean estimation problem. We give a \enquote{no free lunch} result, characterizing the settings (and sample sizes) where PPI++ has provably worse estimation error than using gold-standard labels alone. Specifically, PPI++ will outperform if and only if the correlation between pseudo- and gold-standard is above a certain level that depends on the number of labeled samples ($n$). In some cases our results simplify considerably: For Gaussian data, for instance, the correlation must be at least $1/\sqrt{n - 2}$ in order to see improvement. More broadly, by providing exact non-asymptotic expressions for the variance of PPI++ under sample splitting, we aim to empower practitioners to transparently reason about the benefits of PPI++ in specific applications. In experiments, we illustrate that our theoretical findings hold on real-world datasets.

We examine the concentration of uniform generalization errors around their expectation in binary linear classification problems via an isoperimetric argument. In particular, we establish Poincaré and log-Sobolev inequalities for the joint distribution of the output labels and the label-weighted input vectors, which we apply to derive concentration bounds. The derived results improve upon existing bounds obtained from general unbounded empirical processes, as well as that tailored specifically to logistic regression. In asymptotic analysis, we also show that almost sure convergence of uniform generalization errors to their expectation occurs in very broad settings, such as proportionally high-dimensional regimes. Using this convergence, we establish uniform laws of large numbers under dimension-free conditions.

This paper considers the estimation of quantiles via a smoothed version of the stochastic gradient descent (SGD) algorithm. By smoothing the score function with a bandwidth tied to the learning rate, we obtain estimates that are monotone in the quantile level at every iteration, while retaining the memory and computational efficiency required for streaming data. We establish non-asymptotic tail probability bounds for the smoothed estimate with and without Polyak-Ruppert averaging, which are sub-exponential with a multi-regime structure. For the averaged estimate we further derive a Bahadur representation that is uniform in the quantile level and across coordinates, and a resulting Gaussian approximation by the maximum of Brownian bridges, with the dimension $p$ allowed to grow exponentially in the sample size. This yields simultaneous inference across coordinates and quantile levels. As an alternative that avoids estimating the sparsity function, we propose an online multiplier bootstrap that preserves monotonicity, runs in a single pass and is asymptotically valid. Extending the theory to a localized recursion, we obtain online nonparametric conditional quantile estimates with uniform bands over design points and quantile levels. Simulations confirm accurate finite-sample coverage, and we illustrate the method on conditional value-at-risk curves.

In the context of multivariate nonparametric regression with missing covariates, we propose Pattern Embedded Neural Networks (PENNs), which can be applied in conjunction with any existing imputation technique. In addition to a neural network trained on the imputed data, PENNs pass the vectors of observation indicators through a second neural network to provide a compact representation. The outputs are then combined in a third neural network to produce final predictions. Our main theoretical result exploits an assumption that the observation patterns can be partitioned into cells on which the Bayes regression function behaves similarly, and belongs to a compositional Hölder class. It provides a finite-sample excess risk bound that holds for an arbitrary missingness mechanism, and in combination with a complementary minimax lower bound, demonstrates that our PENN estimator attains in typical cases the minimax rate of convergence as if the cells of the partition were known in advance, up to a poly-logarithmic factor in the sample size. Numerical experiments on simulated, semi-synthetic and real data confirm that the PENN estimator consistently improves, often dramatically, on standard neural networks without pattern embedding. Code to reproduce our experiments, as well as a tutorial on how to apply our method, is publicly available.

Ensuring safe behavior in reinforcement learning (RL) is challenging when safety constraints are implicit and cannot be densely measured. In many settings, supervision is limited to coarse approvals or rejections of whole trajectories (e.g., whether a rollout remained within an unknown safety threshold). We propose TraCeS (Trajectory-based Constraint Estimation for Safety), a method for learning per-timestep violation credit from such sparse trajectory-level labels. TraCeS trains a sequential violation estimator whose per-step credits factorize the predicted probability that a trajectory has not yet violated the constraint, and integrates this learned signal into constrained policy optimization. The method requires neither a known cost function nor a known threshold, and remains compatible with standard continuous-control algorithms. We provide a theoretical analysis of the approximation gap introduced by the learning objective, and demonstrate empirically that TraCeS improves constraint satisfaction and feedback efficiency over baselines across multiple continuous-control benchmarks, including long-horizon tasks and settings with noisy or inconsistent labels.

We investigate the statistical properties of Temporal Difference (TD) learning with Polyak-Ruppert averaging, arguably one of the most widely used algorithms in reinforcement learning, for the task of estimating the parameters of the optimal linear approximation to the value function. Assuming independent samples, we make three theoretical contributions that improve upon the current state-of-the-art results: (i) we establish refined high-dimensional Berry-Esseen bounds over the class of convex sets, achieving faster rates than the best known results, and (ii) we propose and analyze a novel, computationally efficient online plug-in estimator of the asymptotic covariance matrix; (iii) we derive sharper high probability convergence guarantees that depend explicitly on the asymptotic variance and hold under weaker conditions than those adopted in the literature. These results enable the construction of confidence regions and simultaneous confidence intervals for the linear parameters of the value function approximation, with guaranteed finite-sample coverage. We demonstrate the applicability of our theoretical findings through numerical experiments.

We present a finite-sample analysis of decentralized learning in two-player zero-sum matrix games and stochastic games, with a focus on best-response-based learning algorithms. In matrix games, the learning algorithm is payoff-based and symmetric: each player updates its policy using only its own payoff observations, incrementally moving toward an estimated smoothed best response to the opponent's latest policy. For stochastic games, we build on this matrix-game primitive to develop a learning algorithm called value iteration with smoothed best response (VI-SBR), which combines smoothed-best-response learning in induced matrix games with a decentralized, model-free approximation of minimax value iteration. We establish finite-sample guarantees in both settings. For matrix games, our results imply a sample complexity of $\mathcal{O}(ε^{-1})$ for finding an $ε$-Nash distribution and, with explicit exploration, $\tilde{\mathcal{O}}(ε^{-8})$ for finding an $ε$-Nash equilibrium. For stochastic games, we prove that the exploration-enhanced VI-SBR algorithm achieves a sample complexity of $\tilde{\mathcal{O}}(ε^{-8})$ for finding an $ε$-Nash equilibrium. Technically, our analysis develops a coupled Lyapunov-drift framework. This framework simultaneously handles stochastic iterative algorithms with multiple interacting stochastic iterates, the non-zero-sum auxiliary games generated by independently updated value functions, and the time-inhomogeneous Markovian noise induced by time-varying policies. The resulting tools may be useful more broadly for analyzing learning algorithms with coupled stochastic iterates and nonstationary sampling processes.

This paper establishes the generalization error of pooled min-$\ell_2$-norm interpolation in transfer learning, where data from diverse distributions are available. Min-norm interpolators arise naturally as implicit regularized limits of modern machine learning algorithms. Prior work has characterized their out-of-distribution risk when samples from the test distribution are unavailable during training. In many applications, however, limited test samples may be available at training time, yet properties of min-norm interpolation in this regime remain poorly understood. We address this gap by characterizing the bias and variance of pooled min-$\ell_2$-norm interpolation under both covariate shift and model shift. Our results yield several important implications. In certain cases under model shift, we show that adding data always hurts when the signal-to-noise ratio (SNR) is low. At higher SNR levels, transfer learning is beneficial provided the shift-to-signal ratio falls below a threshold that we characterize explicitly. Under covariate shift, we find that when the source sample size is small relative to the dimension, greater heterogeneity between domains reduces risk, and vice versa. While our model shift results are initially established for Gaussian designs, we extend them to more general designs through a universality argument. To illustrate the broader applicability of our technical tools beyond interpolation learning, we characterize the risk of a bias-corrected estimator that uses the pooled interpolator as an initialization and corrects the resulting bias with target data. On the technical side, we develop a novel anisotropic local law and a Lindeberg-swapping argument, yielding tools that may be of independent interest in random matrix theory and universality analysis. Finally, we supplement our theory with simulations demonstrating the finite-sample efficacy of our results.

We endeavour to estimate numerous multi-dimensional means of various probability distributions on a common space based on independent samples. Our approach involves forming estimators through convex combinations of empirical means derived from these samples. We introduce two strategies to find appropriate data-dependent convex combination weights: a first one employing a testing procedure to identify neighbouring means with low variance, which results in a closed-form plug-in formula for the weights, and a second one determining weights via minimization of an upper confidence bound on the quadratic risk. Through theoretical analysis, we evaluate the improvement in quadratic risk offered by our methods compared to the empirical means. Our analysis focuses on a dimensional asymptotics perspective, showing that our methods asymptotically approach an oracle (minimax) improvement as the effective dimension of the data increases. We demonstrate the efficacy of our methods in estimating multiple kernel mean embeddings through experiments on both simulated and real-world datasets.

Analyzing the behavior of complex interdependent networks requires complete information about the network topology and the interdependent links across networks. For many applications such as critical infrastructure systems, understanding network interdependencies is crucial to anticipate cascading failures and plan for disruptions. However, data on the topology of individual networks are often publicly unavailable due to privacy and security concerns. Additionally, interdependent links are often only revealed in the aftermath of a disruption as a result of cascading failures. We propose a scalable nonparametric Bayesian approach to reconstruct the topology of interdependent infrastructure networks from observations of cascading failures. Metropolis-Hastings algorithm coupled with the infrastructure-dependent proposal are employed to increase the efficiency of sampling possible graphs. Results of reconstructing a synthetic system of interdependent infrastructure networks demonstrate that the proposed approach outperforms existing methods in both accuracy and computational time. We further apply this approach to reconstruct the topology of one synthetic and two real-world systems of interdependent infrastructure networks, including gas-power-water networks in Shelby County, TN, USA, and an interdependent system of power-water networks in Italy, to demonstrate the general applicability of the approach.

The empirical risk minimization approach to data-driven decision making requires access to training data drawn under the same conditions as those that will be faced when the decision rule is deployed. However, in a number of settings, we may be concerned that our training sample is biased in the sense that some groups (characterized by either observable or unobservable attributes) may be under- or over-represented relative to the general population; and in this setting empirical risk minimization over the training set may fail to yield rules that perform well at deployment. We propose a model of sampling bias called conditional $Γ$-biased sampling, where observed covariates can affect the probability of sample selection arbitrarily much but the amount of unexplained variation in the probability of sample selection is bounded by a constant factor. Applying the distributionally robust optimization framework, we propose a method for learning a decision rule that minimizes the worst-case risk incurred under a family of test distributions that can generate the training distribution under $Γ$-biased sampling. We apply a result of Rockafellar and Uryasev to show that this problem is equivalent to an augmented convex risk minimization problem. We give statistical guarantees for learning a model that is robust to sampling bias via the method of sieves, and propose a deep learning algorithm whose loss function captures our robust learning target. We empirically validate our proposed method in a case study on prediction of mental health scores from health survey data and a case study on ICU length of stay prediction.

The problem of linear predictions has been extensively studied for the past century under pretty generalized frameworks. Recent advances in the robust statistics literature allow us to analyze robust versions of classical linear models through the prism of Median of Means (MoM). Combining these approaches in a piecemeal way might lead to ad-hoc procedures, and the restricted theoretical conclusions that underpin each individual contribution may no longer be valid. To meet these challenges coherently, in this study, we offer a unified robust framework that includes a broad variety of linear prediction problems on a Hilbert space, coupled with a generic class of loss functions. Notably, we do not require any assumptions on the distribution of the outlying data points ($\mathcal{O}$) nor the compactness of the support of the inlying ones ($\mathcal{I}$). Under mild conditions on the dual norm, we show that for misspecification level $ε$, these estimators achieve an error rate of $O(\max\left\{|\mathcal{O}|^{1/2}n^{-1/2}, |\mathcal{I}|^{1/2}n^{-1} \right\}+ε)$, matching the best-known rates in literature. This rate is slightly slower than the classical rates of $O(n^{-1/2})$, indicating that we need to pay a price in terms of error rates to obtain robust estimates. Additionally, we show that this rate can be improved to achieve so-called "fast rates" under additional assumptions.

In many applications, researchers are interested in the direct and indirect causal effects of a treatment or exposure on an outcome of interest. Mediation analysis offers a rigorous framework for identifying and estimating these causal effects. For binary treatments, efficient estimators for the direct and indirect effects are presented by Tchetgen Tchetgen and Shpitser (2012) based on the influence function of the parameter of interest. These estimators possess desirable properties such as multiple-robustness and asymptotic normality while allowing for slower than root-n rates of convergence for the nuisance parameters. However, in settings involving continuous treatments, these influence function-based estimators are not readily applicable without making strong parametric assumptions. In this work, utilizing a kernel smoothing approach, we propose an estimator suitable for settings with continuous treatments inspired by the influence function-based estimation strategy. Our proposed approach employs cross-fitting, relaxing the smoothness requirements on the nuisance functions and allowing them to be estimated at slower rates than the target parameter. Additionally, similar to influence function-based estimators, our proposed estimator is multiply robust and asymptotically normal, allowing for inference in settings where parametric assumptions may not be justified.

Inference for factor models is often hampered by the lack of tractable and accurate variance estimates, which can materially distort downstream analyses. In practice, uncertainty in the residual covariance matrix is frequently either ignored or addressed through computationally intensive resampling methods that tend to be unstable. This paper develops a unified analytical framework for inference in exploratory factor analysis under several widely used extraction rules, including least-squares, principal-factor, iterative principal-component, alpha, and image factoring. By treating these estimators as implicitly defined functions of the sample covariance matrix, we derive closed-form Jacobians that translate perturbations in the covariance matrix into changes in the resulting factor solutions. Combined with the delta method and consistent estimators of the sample covariance matrix, the proposed approach yields standard errors that are straightforward to compute and remain valid under non-Gaussianity, heteroskedasticity, and serial or cross-sectional dependence. Simulation evidence confirms that the analytical standard errors accurately capture finite-sample variability while avoiding both the instability of bootstrap procedures and the restrictive assumptions underlying Fisher information-based inference. An application to a factor-augmented structural vector autoregressive (SVAR) model further demonstrates how accounting for this source of uncertainty can substantially affect impulse-response inference. Taken together, the results provide a practical and general tool for propagating estimation uncertainty in settings where factor extraction serves as an intermediate step.

We revisit the problem of estimating $k$ linear regressors with self-selection bias in $d$ dimensions with the maximum selection criterion, as introduced by Cherapanamjeri, Daskalakis, Ilyas, and Zampetakis [CDIZ23, STOC'23]. Our main result is a $\mathrm{poly}(d, k, 1/\varepsilon) + (k \log k)^{O(k)}$ time algorithm for this problem that improves upon the running time of the algorithms by Cherapanamjeri, Daskalakis, Ilyas, and Zampetakis [CDIZ23] and Gaitonde and Mossel [GM24, arXiv]. We achieve this by providing the first local convergence algorithm for self-selection, thus resolving one of the main open questions of Cherapanamjeri, Daskalakis, Ilyas, and Zampetakis [CDIZ23]. To obtain this algorithm, we reduce self-selection to a seemingly unrelated statistical problem called estimation under coarsening [FKKT21, COLT'21]. Coarsening occurs when one does not observe the exact value of the sample but only some set (from a partition of the sample space) containing the exact value. Inference from coarse samples arises in various real-world applications, including rounding by humans and algorithms, limited precision of instruments, and lag in multi-agent systems. The coarse estimation problem arising in our reduction is induced by a non-convex partition, whereas previous works on coarsening exclusively studied convex partitions. The resulting estimation algorithm relies on the geometry of the self-selection problem to bypass non-convexity. This geometric approach, in turn, enables us to overcome the limitations of previous analytic approaches and could have applications for designing efficient algorithms for other latent-variable problems.

We propose a generalization of the synthetic control methods to the setting with dynamic treatment effects, in which each unit receives multiple treatments sequentially, according to an adaptive policy that depends on a latent, endogenously time-varying confounding state. Under a low-rank latent factor model assumption, which admits linear time-varying and time-invariant dynamic triangular systems as special cases, we develop an identification strategy for any unit-specific mean outcome under any sequence of interventions. Our method, which we term synthetic blips, is a backward induction process in which the blip effect of a treatment at each period for a target unit is recursively expressed as a linear combination of the blip effects of other units that received the designated treatment, avoiding the combinatorial donor requirements of naive synthetic control extensions. We provide easy-to-implement estimation algorithms that yield consistent estimators. Using unique Korean firm-level panel data, we estimate individualized dynamic treatment effects and optimal allocation rules in the context of financial support for exporting firms.