Treatment Effect

We study the problem of estimating the effect function for a continuous treatment, which maps each treatment value to a population-averaged outcome. A central challenge in this setting is confounding: treatment assignment often depends on covariates, creating selection bias that makes direct regression of the response on treatment unreliable. To address this issue, we propose a two-stage kernel ridge regression method. In the first stage, we learn a model for the response as a function of both treatment and covariates; in the second stage, we use this model to construct pseudo-outcomes that correct for distribution shift, and then fit a second model to estimate the treatment effect. Although the response varies with both treatment and covariates, the induced effect function obtained by averaging over covariates is typically much simpler, and our estimator adapts to this structure. Our optimal learning bounds are achieved without estimating the conditional treatment density, thereby bypassing a major bottleneck in existing methods. Furthermore, we introduce a fully data-driven model selection procedure that achieves provable adaptivity to both the unknown degree of overlap and the spectral decay of the underlying kernel.

Many decision-making problems require ranking individuals by their treatment effects rather than estimating the exact effect magnitudes. Examples include prioritizing patients for preventive care interventions, or ranking customers by the expected incremental impact of an advertisement. Surprisingly, while causal effect estimation has received substantial attention in the literature, the problem of directly learning *rankings of treatment effects* has largely remained unexplored. In this paper, we introduce *Rank-Learner*, a novel two-stage learner that directly learns the ranking of treatment effects from observational data. We first show that naive approaches based on precise treatment effect estimation solve a harder problem than necessary for ranking, while our *Rank-Learner* optimizes a pairwise learning objective that recovers the true treatment effect ordering, without explicit CATE estimation. We further show that our *Rank-Learner* is Neyman-orthogonal and thus comes with strong theoretical guarantees, including robustness to estimation errors in the nuisance functions. In addition, our *Rank-Learner* is model-agnostic, and can be instantiated with arbitrary machine learning models (e.g., neural networks). We demonstrate the effectiveness of our method through extensive experiments where *Rank-Learner* consistently outperforms standard CATE estimators and non-orthogonal ranking methods. Overall, we provide practitioners with a new, orthogonal two-stage learner for ranking individuals by their treatment effects.

We present improved bounds for estimating discrete probability distributions under the $\ell_{\infty}$ norm. These include minimax bounds in expectation and high-probability tail bounds. We resolve some of the open questions posed in Kontorovich and Painsky (JMLR, 2025) --- including a fully empirical version of the tightest risk bound they presented and identifying the form of the worst-case extremal distribution. Encouraging empirical results are reported as well.

As network data has become ubiquitous in the sciences, there has been growing interest in network models whose structure is driven by latent node-level variables in a (typically low-dimensional) latent geometric space. These "latent positions" are often estimated via embeddings, whereby the nodes of a network are mapped to points in Euclidean space so that "similar" nodes are mapped to nearby points. Under certain model assumptions, these embeddings are consistent estimates of the latent positions, but most such results require the embedding dimension to be chosen correctly. Methods for choosing the embedding dimension have been studied extensively, but little is known about the behavior of embeddings when the dimension is misspecified. In this work, we provide a theoretical description of the effects of dimension misspecification under the random dot product graph, a class of latent space network models that includes several widely-used network models, most notably the stochastic blockmodel, as special cases. We show that when the dimension is chosen too large, consistent estimation still holds, albeit at a slower rate than when the embedding dimension is chosen correctly. On the other hand, when the dimension is chosen too small, there is a fundamental estimation error lower bound that need not go to zero in the large-network limit. A range of synthetic data experiments support our theoretical results. Our main technical result, which may be of independent interest, is a generalization of earlier work in random matrix theory showing that all non-signal eigenvectors of a low-rank matrix subject to additive noise are delocalized.

This paper studies the use of *kernel density estimation* (KDE) for linear algebraic tasks involving the *kernel matrix* of a collection of $n$ data points in $\mathbb{R}^d$. In particular, we improve upon the best existing algorithms for computing the following up to $(1+\varepsilon)$ relative error for a Gaussian kernel matrix and other kernels: matrix-vector products, matrix-matrix products, the spectral norm, and sum of all entries. The runtimes of our algorithms depend linearly on the dimension $d$, sub-quadratically in the number of points $n$, and polynomially on the target error $\varepsilon$. Importantly, the dependence on $n$ in each case is far lower when accessing the kernel matrix through KDE queries as opposed to reading individual entries. Our improvements over existing best algorithms (particularly those of [Backurs et al. ICML `21]) for these tasks reduce the polynomial dependence on $\varepsilon$, and additionally decrease the dependence on $n$ in the case of computing the sum of all entries of the kernel matrix. For example, we reduce the power of $1/\epsilon$ from $\approx 7.7$ to $\approx 3.2$ for a $1-\varepsilon$ relative error estimation of the spectral norm of a Gaussian kernel matrix. We complement our upper bounds with several lower bounds for related problems, which provide (conditional) quadratic time hardness results and additionally hint at the limits of KDE based approaches for the problems we study.

Causal inference in modern large-scale systems faces growing challenges, including high-dimensional covariates, multi-valued treatments, massive observational (OBS) data, and limited randomized controlled trial (RCT) samples due to cost constraints. We formalize treatment-induced structural non-overlap and show that, under this regime, commonly used weighted fusion methods provably fail to satisfy randomized identifying restrictions.To address this issue,we propose a constrained joint estimation framework that minimizes observational risk while enforcing causal validity through orthogonal experimental moment conditions. We further show that structural non-overlap creates a feasibility obstruction for moment enforcement in the original covariate space.We also derive a penalized primal–dual algorithm that jointly learns representations and predictors, and establish oracle inequalities decomposing error into overlap recovery, moment violation, and statistical terms.Extensive synthetic experiments demonstrate robust performance under varying degrees of non-overlap. A large-scale ride-hailing application shows that our method achieves substantial gains over existing baselines, matching the performance of models trained with significantly more RCT data.

Estimating Heterogeneous Treatment Effects (HTE) from observational data is essential in fields such as healthcare and policy-making, where randomized experiments are often impractical. While representation learning-based methods have shown promise, recent studies suggest that tree-based approaches may offer superior performance on tabular data, particularly in the presence of uninformative features. We introduce **GRA**dient-based **C**ausal tree **E**nsembles (GRACE), a novel tree-based architecture for HTE estimation that incorporates multi-way, oblique, and soft splits, enabling end-to-end training via backpropagation. GRACE can be seamlessly integrated into existing models as a replacement for fully-connected neural network layers. Across diverse benchmarks involving binary and non-binary treatment settings, GRACE consistently surpasses neural network and tree-based baselines, often by a substantial margin. We further analyze GRACE as an extension of fully-connected neural network layers and conduct ablation studies to isolate and quantify the contribution of each architectural component to the improvement in performance. These results position GRACE as a promising new foundation for flexible, robust, and accurate HTE estimation. GRACE code is available at https://github.com/ysk-kano/GRACE

Causal effect estimation is a fundamental task in many scientific fields. Selecting appropriate covariates for adjustment is crucial for obtaining unbiased causal effects. However, most existing methods either rely on learning the global causal structure, assume the absence of latent variables, or impose the pretreatment assumption-restricts covariates to those unaffected by the treatment or outcome. These assumptions are often unrealistic in real-world scenarios, and global structure learning can be computationally intensive and inefficient. To address these challenges, we first characterize the local existence boundary of adjustment sets for causal effect estimation. Based on this characterization, we develop a novel local learning method for covariate selection in nonparametric causal effect estimation. This method accommodates the presence of latent variables and eliminates the need for the pretreatment assumption. We prove that the proposed method is both sound and complete under standard assumptions. Its effectiveness is validated through extensive experiments on both synthetic and real-world datasets.

Foundation models based on prior-data fitted networks (PFNs) have shown strong empirical performance in causal inference by framing the task as an in-context learning problem. However, it is unclear whether PFN-based causal estimators provide uncertainty quantification that is consistent with classical frequentist estimators. In this work, we address this gap by analyzing the frequentist consistency of PFN-based estimators for the average treatment effect (ATE). (1) We show that existing PFNs, when interpreted as Bayesian ATE estimators, can exhibit prior-induced confounding bias: the prior is not asymptotically overwritten by data, which, in turn, prevents frequentist consistency. (2) As a remedy, we suggest employing a calibration procedure based on a one-step posterior correction (OSPC). We show that the OSPC helps to restore frequentist consistency and can yield a semi-parametric Bernstein-von Mises theorem for calibrated PFNs (i.e., both the calibrated PFN-based estimators and the classical semi-parametric efficient estimators converge in distribution with growing data size). (3) Finally, we implement OSPC through tailoring martingale posteriors on top of the PFNs. In this way, we are able to recover functional nuisance posteriors from PFNs, required by the OSPC. In multiple (semi-)synthetic experiments, PFNs calibrated with our martingale posterior OSPC produce ATE uncertainty that (i) asymptotically matches frequentist uncertainty and (ii) is well calibrated in finite samples in comparison to other Bayesian ATE estimators.

We present a regression-adjustment framework designed for the estimation of longitudinal treatment effects in randomized experiments under static regimes. While regression-adjustment methods are useful for variance reduction in randomized experiments by using pre-treatment covariates, they usually focus only on average effects, from which we cannot obtain valuable insights into when the effects appear and how long they continue. To address this issue, we consider intermediate outcomes and evolving post-treatment covariates over time, and we represent such dynamic trajectories using transition kernels. Furthermore, we establish the asymptotic normality and the semiparametric efficiency bound for our estimator, enabling more powerful statistical inference. Simulation studies and empirical analysis using A/B test data from a streaming platform in Japan show the practical advantages of our method.

Offline contextual pricing often relies on logged observational data, but faces challenges from distributional shifts between training and deployment environments. Distributionally robust optimization (DRO) provides a principled approach to off-policy evaluation and learning (OPE/L). However, existing methods are mostly limited to discrete actions. Recent work has explored DRO for continuous treatments using inverse propensity weighting (IPW), while such IPW-based estimators can be sensitive to the convergence rate of propensity score estimates, particularly when estimated nonparametrically, which may lead to larger estimation errors and regret. In this work, we develop a doubly robust (DR) framework for distributionally robust OPE/L in continuous pricing settings. For evaluation, we propose a localized DR estimator that addresses the computational challenges of worst-case expectations by fitting only a small number of regressions, comparable to standard non-robust DR, while achieving semiparametric efficiency under mild product rate conditions. For learning, we leverage the inherent smoothness of demand noise to handle pricing-specific discontinuities in revenue outcomes (e.g., threshold-based purchase decisions), establishing a finite-sample regret bound of $\tilde{\mathcal{O}}_p(T^{-s/(2s+1)})$ for smoothness orders $s=1,2$. This bound improves upon existing regret rates in existing DRO-based off-policy learning (OPL) for continuous treatments. Extensive experiments under various levels of distribution shift validate our proposed framework.

Logistic regression is widely used in applications; however, when the dimension scales with the sample size, theory reveals that the asymptotic behavior of common M-estimators depends on nonzero bias and variance factors, which are functions of the signal strength. To leverage the theory to design valid statistical methodologies, it is essential to obtain accurate estimates of the signal strength. In this work, we utilize a data-splitting strategy to efficiently estimate the signal strength. To alleviate issues caused by separable data, we analyze the exact asymptotics of an M-estimator with a data-driven, non-decomposable regularizer that adapts to the true covariance structure. We justify the validity of our method through both theoretical analysis and numerical experiments.

We study bilinear bandits with partially observable features on both the user and item sides. In each of $T$ rounds, the learner selects a user--item pair and observes only the reward for the chosen pair. The reward model is bilinear in the user and item features with an unknown parameter matrix. Existing work commonly reduces this problem to a linear bandit via Kronecker-product features, increasing dimensionality and losing bilinear structure. We propose \texttt{BiRoLF}, an algorithm robust to latent features, which directly leverages the bilinear structure without such linearization. It augments observed feature spaces on both sides with orthogonal complement bases and employs doubly robust (DR) estimation to impute rewards for unselected pairs, constructing matrix-valued pseudo-rewards. We estimate the effective parameter using a bilinear DR-Lasso estimator, which promotes sparsity in components orthogonal to observed features. \texttt{BiRoLF} achieves a $\tilde{O}(\sqrt{(d_x + d_{h_x})(d_y + d_{h_y}) T})$ regret bound, where $d_x$ and $d_y$ are observable feature dimensions, and $d_{h_x}$ and $d_{h_y}$ denote effective orthogonal-complement dimensions. By exploiting the induced block-diagonal Gram structure, \texttt{BiRoLF} performs an exact blockwise update that preserves the full DR-Lasso objective. Numerical experiments show strong regret performance and computational gains.

Comparative evaluation of *multiple* dynamic treatment policies is essential for healthcare and policy decisions, yet conventional longitudinal causal inference methods estimate each in *isolation*, preventing information sharing across counterfactuals. We demonstrate that this separate estimation paradigm induces a structurally uncontrolled second-order bias, inflating finite-sample variance even after standard debiasing with *longitudinal targeted maximum likelihood estimation* (LTMLE). To address this, we propose a policy-aware reparameterization of *Iterative Conditional Expectation* (ICE) Q-functions that enables joint estimation through shared representations. We implement this approach in the **Policy-Encoded Q Network (PEQ-Net)**, an architecture centered on a shared policy encoder. The encoder is trained using kernel mean embeddings, ensuring that the learned representation space reflects population-level policy dissimilarities. After applying an LTMLE correction step, we prove this design imposes a structural constraint on the second-order remainder, thereby stabilizing finite-sample variance. Experiments on semi-synthetic datasets demonstrate that PEQ-Net consistently outperforms existing ICE-based methods, achieving substantial reductions in root-mean-square error, particularly when evaluating closely related policies.

Traditional instrumental variable (IV) estimators cannot accommodate more treatments than instruments, a limitation that is critical for high-dimensional, unstructured data like clinical treatment pathways. Current practice—applying unsupervised dimension reduction before IV estimation—suffers from substantial omitted treatment bias because the representation learning step ignores the instrument. We propose a novel framework that constructs treatment representations by explicitly incorporating instrumental variables. We prove that this instrument-guided approach ensures the identification of optimal outcome-prediction directions even with limited instruments. Validation on large-scale, semi-synthetic clinical data derived from a major hospital, along with other simulations, shows that our approach significantly outperforms conventional two-stage methods.

Amortized causal inference via Prior-data Fitted Networks (PFNs) has emerged as a promising paradigm, enabling zero-shot estimation of causal effects without the need for dataset-specific model tuning. However, the principled effectiveness of unified pre-training across general interventional regimes remains an underexplored question. In this paper, we investigate interventions on subsets of variables within Structural Causal Models (SCMs) and identify a fundamental theoretical limitation of current pre-training approaches. Theoretically, we prove that a single observational SCM induces an exponentially large space of interventional distributions, resulting in a phenomenon we term prior uncoverage. Consequently, this uncoverage yields a mismatch between the learned meta-prior and the true grounding prior, leading to unavoidable posterior inconsistency and estimation bias. To address this, we posit that fine-tuning is a fundamental necessity and propose a target-specific strategy named Point-Wise Interventional Fine-tuning (PWF), enabling the local generalization property. We further scale this approach via Meta-Sampling Fine-tuning (MSF) from a budgeted active learning perspective, thereby achieving uniform generalization on any interventional distribution.

Randomized experiments (or A/B tests) are widely used to evaluate interventions in dynamic systems such as recommendation platforms, marketplaces, and digital health. In these settings, interventions affect both current and future system states, so estimating the global average treatment effect (GATE) requires accounting for temporal dynamics, which is especially challenging in the presence of nonstationarity; existing approaches suffer from high bias, high variance, or both. In this paper, we address this challenge via the novel Truncated Policy Gradient (TPG) estimator, which replaces instantaneous outcomes with short-horizon outcome trajectories. The estimator admits a policy gradient interpretation: it is a truncation of the first-order approximation to the GATE, yielding provable reductions in bias and variance in nonstationary Markovian settings. We further establish a central limit theorem for the TPG estimator and develop a consistent variance estimator that remains valid under nonstationarity with single-trajectory data. We validate our theory with two real-world case studies. The results show that relative to existing approaches, a well-calibrated TPG estimator can achieve a favorable balance between bias and variance in nonstationary settings, highlighting the value of the policy-gradient perspective for designing effective estimators under complex dynamics.

Treatment responder classification seeks to learn a rule to classify individuals who will benefit from the treatment. This paper studies a new scenario in treatment responder classification when abstention is allowed, i.e., practitioners can opt out of making uncertain classification on some individuals for further investigation. By revealing the implicit relation between causal misclassification risk with abstention and Conditional Value at Risk (CVaR), we develop a doubly robust method named TRECA to learn the classification rule under loose convergence conditions on nuisance parameters, and further extend it to deal with possible violation on key assumptions such as monotonicity and unconfoundedness. Rigorous theories and extensive experiments on two real-world datasets demonstrate the theoretical and experimental guarantee on our methods in learning treatment responders classification rules with low regret at the cost of limited abstention.

Mean Field Variational Inference (MFVI) is widely understood to underestimate posterior variance. By analysing conjugate Bayesian linear regression, we show that this characterisation is incomplete: while MFVI underestimates the variance in parameter space, it can overestimate the predictive variance compared to the exact posterior. We show that if the MFVI posterior underestimates predictive variances in some directions, it necessarily overestimates them in others. Crucially, this overestimation occurs in directions where the training data concentrates. This leads to the surprising result that, for a test point drawn from the training distribution, MFVI's expected predictive variance exceeds that of the exact posterior. We demonstrate a pathological case of this effect, where the MFVI posterior fails to reduce predictive variance compared to the prior on i.i.d. data. We connect these results to the Cold Posterior Effect, arguing that varying the temperature can correct this overestimation, yielding predictions closer to those of the exact posterior. We validate our theory on synthetic and real-world regression tasks.

Estimating heterogeneous treatment effects is central to data-driven decision-making, yet industrial applications often face a fundamental tension between limited randomized controlled trial (RCT) budgets and abundant but biased observational data (OBS) collected under historical targeting policies. Although observational logs offer the advantage of scale, they may suffer from severe policy-induced imbalance and overlap violations, rendering standalone estimation unreliable. We propose a \textit{budgeted active experimentation} framework that iteratively collects informative randomized samples for causal effect estimation via active sampling. By leveraging observational signals, we develop an acquisition function targeting uplift estimation uncertainty, domain discrepancy, and overlap deficits to select the most informative units for randomized experiments. We establish finite-sample deviation bounds, asymptotic normality via martingale CLTs, and minimax lower bounds showing near-optimality in the linear representation setting. Experiments on synthetic datasets support our theoretical findings, and further extensions to industrial neural network-based uplift modeling scenarios show that active sampling can improve sample efficiency over random sampling under limited RCT budgets.