Covariate Shift

Confidence calibration for classification models is vital in safety-critical decision-making scenarios and has received extensive attention. General confidence calibration methods assume training and test data are independent and identically distributed, limiting their effectiveness under covariate shifts. Previous calibration methods under covariate shift struggle with class-wise or canonical calibrations and often rely on unstable importance weighting when density ratios are large or unbounded. Given the above limitations, this paper rethinks confidence calibration under covariate shifts. First, we derive a necessary and sufficient condition for confidence calibration under covariate shifts, named Expectation consistency condition, which reveals covariate shifts do not necessarily lead to uncalibrated confidence and provides a weaker condition for confidence calibration than global covariate distribution alignment. Then, utilizing Expectation consistency condition, this paper proposes an unsupervised domain adaptation loss to calibrate confidence of the target domain, named Expectation consistency loss (ECL), which is compatible with canonical calibration, class-wise calibration, and top-label calibration. Third, we prove that computing ECL loss has the same sample complexity as Expected Calibration Error (ECE) and provide a theoretically grounded mini-batch trainable scheme for ECL loss. Finally, we validate the effectiveness of our method on both simulated and real-world covariate shift datasets.

A unified framework for learning under covariate shift is presented, in which a constrained density-ratio network approximates the Radon-Nikodym derivative $r^\star = dP/dQ$ and feeds an anytime PAC-Bayes generalization certificate. A change-of-measure identity decomposes the gap between target risk and importance-weighted source risk into a ratio-bias term governed by $\|r_θ- r^\star\|_{L^2(Q)}$ and a generalization-gap term governed by the variability of the weighted loss. Normalization and moment-matching identities are enforced as hard integral constraints through an augmented-Lagrangian scheme, with a second-moment penalty controlling the effective sample size. PAC-Bayes is instantiated on the weighted risk in a fixed-time regime that yields Bernoulli-KL bounds, identifies the network-weighted Gibbs posterior as the unique KL-regularized minimizer, and quantifies stability under $L^2(Q)$ perturbations of the learned ratio, and is then strengthened by geometric peeling to an anytime certificate uniform in $t \geq t_{\min}$. A pre-registered two-campaign protocol combining a patch test against analytic ground truth with a real-data deployment validates the framework: the network produces calibrated ratios, reduces target $0/1$ loss against unweighted ERM and classical direct ratio-estimation baselines, and attains the anytime certificate. A single fixed-time coverage failure is recorded, with per-split coverage aligning one-to-one with the magnitude of the label shift, confirming that the covariate-only assumption is operationally tight rather than a defect of the certificate.

We introduce and analyze Target-Induced Loss Tilting (TILT) for unsupervised domain adaptation under covariate shift. It is based on a novel objective function that decomposes the source predictor as $f+b$, fits $f+b$ on labeled source data while simultaneously penalizing the auxiliary component $b$ on unlabeled target inputs. The resulting fit $f$ is deployed as the final target predictor. At the population level, we show that this target-side penalty implicitly induces relative importance weighting at the population level, but in terms of an estimand $b^*_f$ that is self-localized to the current error, and remains uniformly bounded for any source-target pair (even those with disjoint supports). We prove a general finite-sample oracle inequality on the excess risk, and use it to give an end-to-end guarantee for training with sparse ReLU networks. Experiments on controlled regression problems and shifted CIFAR-100 distillation show that TILT improves target-domain performance over source-only training, exact importance weighting, and relative density-ratio baselines, with a stable dependence on the regularization parameter.

As synthetic images become increasingly realistic, reliable synthetic image detection techniques are of pressing need to prevent their misuse. Despite satisfactory in-distribution performance, deep neural network-based synthetic image detectors (SIDs) lack reliability in deployment and often fail in the presence of common covariate shifts, resulting in poor detection accuracy. To avoid the risk caused by potential errors, we adopt a selective classification (SC) strategy by allowing SIDs to abstain from making low confidence predictions. For practicality, we focus on post-hoc methods which perform confidence estimation on a given SID without retraining. However, we show that conventional logit-based confidence score functions (CSFs) exhibit pathological behavior under covariate shifts, leading to SC performance close to or even worse than random guessing. To address this, we propose a simple yet effective SC framework for Reliable Synthetic Image Detection (ReSIDe). First, we generalize the notion of logits to an SID's intermediate layers from a centroid matching perspective, extending the use of logit-based CSFs to any layer of an SID. Then, we introduce a preference optimization algorithm that aggregates confidence scores extracted from different layers to a final confidence estimate by minimizing an upper bound of the area under the risk-coverage curve (AURC). Extensive experimental results show that ReSIDe significantly boosts the SC performance of various logit-based CSFs under common covariate shifts, achieving up to 69.55% AURC reduction.

Learning parity functions, more general modular addition, is a challenging machine learning task due to its input sensitivity. A recent study substantially scaled modular addition learning in both the number of summands and the modulus. Its key idea is to increase zeros in training sequences, reducing the effective number of summands and thus controlling training difficulty; however, this induces covariate shift between training and test input distributions. This study theoretically and empirically analyzes this side effect and proposes a covariate-shift-free method for modular addition. Specifically, we introduce an auxiliary modulus $Kq$ during training, which reduces wrap-around frequency and problem difficulty while preserving the same input distribution across training and testing. Experiments show strong scalability and sample efficiency: even for large input length $N$, large modulus $q$, and small datasets -- where the sparse method fails to learn -- our method achieves equal or better match accuracy and relaxed $τ$-accuracy. For example, at $N=64$ and $q=974269$, our method trained on 100K samples achieves $97.0\%$ $τ$-accuracy at $τ=0.05$, while the sparse method achieves only $9.5\%$ with the same data size and $93.9\%$ even when extended to 1M samples.

We study long-horizon deployment of a frozen predictor under dynamic covariate shift. A time-domain Poincare inequality first reduces temporal risk volatility to derivative energy. A Jacobian-velocity theorem then supplies the corresponding pathwise control. Given explicit regularity and domination assumptions, the theorem identifies directional tangent energy along the deployment path as the governing quantity. Under low-rank drift, that quantity reduces to directional Jacobian energy in the drift subspace, motivating drift-aligned tangent regularization (DTR) and a matched monitoring proxy. Rather than smoothing the network isotropically, DTR penalizes sensitivity only along estimated drift directions. We validate the theorem-to-method pipeline in four experiments: a synthetic benchmark for the time-domain inequality, a controlled synthetic comparison against isotropic Jacobian regularization, and two frozen-deployment studies on the UCI Air Quality and Tetouan power-consumption datasets. DTR reduces risk volatility and directional gain in the controlled low-rank regime and beats isotropic smoothing there. It also gives validation-selected deployment gains on both real datasets, with the Air Quality subspace estimated from target-orthogonal sensor motion. Moderate drift-subspace misspecification is tolerable while orthogonal misspecification largely removes the benefit.

Conformal prediction (CP) provides powerful, distribution-free prediction sets, but its guarantees rely on the exchangeability of training and test data, which is often violated in practice due to covariate shifts. While weighted conformal prediction (WCP) is designed to handle such shifts, it can suffer from significant undercoverage when the density ratio between the distributions is unbounded and/or must be learned. This is because of both overfitting in learning the density ratio, and high variance in estimating the nonconformity score threshold. To address this, we introduce clipped least-squares importance fitting (CLISF) as a reduced-variance method for density ratio estimation. Specifically, we show that density ratios learned using CLISF, when plugged into WCP, have bounded expected undercoverage. Furthermore, we show that the undercoverage can be corrected by running WCP with a slightly inflated coverage target; crucially, we are able to estimate the required level of inflation from the data. We provide the first theoretical guarantees for weight clipping in conformal inference, achieving dataset-conditional coverage with a sample complexity that does not blow up with the higher moments of the true density ratio -- a key limitation of prior work. We verify our results on real-world benchmarks and synthetic data.

``How long can I live and remain free of cancer?'' is often the first question a patient asks after receiving a cancer diagnosis and treatment. Accurate survival prediction helps alleviate psychological distress and supports risk stratification and personalized treatment planning. Recent survival prediction frameworks have shown strong performance using computed tomography (CT) images. However, variations in imaging acquisition introduce out-of-distribution (OOD) samples caused by covariate shifts that undermine model reliability. Despite this challenge, to our knowledge, no existing benchmark systematically studies OOD detection in cancer survival prediction. To address this gap, we introduce the Cancer sURvival bEnchmark for OOD Detection (CURE-OOD), the first benchmark for systematically evaluating OOD detection in survival prediction under controlled acquisition-induced distribution shifts. CURE-OOD defines scanner-parameter-based training, in-distribution (ID), and OOD test splits across four survival prediction tasks. Our experiments show that covariate shifts notably reduce survival prediction performance. It also shows that mainstream classification-oriented OOD detectors can fail in survival prediction. Finally, we include HazardDev as a simple survival-aware reference baseline for OOD detection. CURE-OOD enables systematic analysis of how distribution shifts affect both downstream survival performance and OOD detectability.

Estimating the prevalence of a category in a population using imperfect measurement devices (diagnostic tests, classifiers, or large language models) is fundamental to science, public health, and online trust and safety. Standard approaches correct for known device error rates but assume these rates remain stable across populations. We show this assumption fails under covariate shift and that multicalibration, which enforces calibration conditional on the input features rather than just on average, is sufficient for unbiased prevalence estimation under such shift. Standard calibration and quantification methods fail to provide this guarantee. Our work connects recent theoretical work on fairness to a longstanding measurement problem spanning nearly all academic disciplines. A simulation confirms that standard methods exhibit bias growing with shift magnitude, while a multicalibrated estimator maintains near-zero bias. While we focus the discussion mostly on LLMs, our theoretical results apply to any classification model. Two empirical applications -- estimating employment prevalence across U.S. states using the American Community Survey, and classifying political texts across four countries using an LLM -- demonstrate that multicalibration substantially reduces bias in practice, while highlighting that calibration data should cover the key feature dimensions along which target populations may differ.

Changes in input distribution can induce shifts in the average predictions of machine learning models. Such prediction shifts may impact downstream business outcomes (e.g. a bank's loan approval rate), so understanding their causes can be crucial. We propose \ours{}: a Shapley value method for attributing prediction shifts to changes in the conditional probabilities of interpretable subgroups of data, where these subgroups are defined by the structure of decision trees. We initially apply this method to single decision trees, providing exact explanations based on conditional probability changes at split nodes. Next, we extend it to tree ensembles by selecting the most explanatory tree and accounting for residual effects. Finally, we propose a model-agnostic variant using surrogate trees grown with a novel objective function, allowing application to models like neural networks. While exact computation can be intensive, approximation techniques enable practical application. We show that \ours{} provides simple, faithful, and near-complete explanations of prediction shifts across model classes, aiding model monitoring in dynamic environments.

Out-of-distribution (OoD) generalization occurs when representation learning encounters a distribution shift. This occurs frequently in practice when training and testing data come from different environments. Covariate shift is a type of distribution shift that occurs only in the input data, while the concept distribution stays invariant. We propose RIA - Regularization for Invariance with Adversarial training, a new method for OoD generalization under convariate shift. Motivated by an analogy to $Q$-learning, it performs an adversarial exploration for counterfactual data environments. These new environments are induced by adversarial label invariant data augmentations that prevent a collapse to an in-distribution trained learner. It works with many existing OoD generalization methods for covariate shift that can be formulated as constrained optimization problems. We develop an alternating gradient descent-ascent algorithm to solve the problem in the context of causally generated graph data, and perform extensive experiments on OoD graph classification for various kinds of synthetic and natural distribution shifts. We demonstrate that our method can achieve high accuracy compared with OoD baselines.

Africa's green industrialization imperative demands reliable infrastructure for monitoring air quality. We present a satellite-reanalysis PM2.5 fusion system trained on 2,068,901 records from 404 monitoring locations in 29 African countries (OpenAQ, 2017-2022), combining LightGBM with leakage-resistant spatial cross-validation and conformal prediction to quantify predictions and their geographic applicability limits. Under 5-fold location-grouped spatial cross-validation, LightGBM achieves RMSE = 30.83 +/- 5.07 ug/m3, MAE = 14.54 +/- 1.66 ug/m3, R2 = 0.134 +/- 0.023, and macro F1 = 0.336 +/- 0.018. This R2 is substantially below random-split benchmarks (>0.90) but reflects true geographic generalisation difficulty rather than model failure. Split conformal prediction targeting 90% marginal coverage reveals severe East Africa degradation (actual PICP = 65.3% vs. nominal 90%), consistent with medium-strength covariate shift (humidity KS = 0.2237, sat_pblh KS = 0.2558). We operationalise these findings through regional reliability flags (High/Medium/Low/Unreliable) and a monitor prioritisation score directing infrastructure expansion toward highest-burden unmonitored populations, directly supporting Africa's green industrial transition and SDGs 3.9, 7.1.2, 9, 11.6.2, and 13.

Accurate forecasting of recovery rates (RR) is central to credit risk management and regulatory capital determination. In many loan portfolios, however, RR modeling is constrained by data scarcity arising from infrequent default events. Transfer learning (TL) offers a promising avenue to mitigate this challenge by exploiting information from related but richer source domains, yet its effectiveness critically depends on the presence and strength of distributional shifts, and on potential heterogeneity between source and target feature spaces. This paper introduces FT-MDN-Transformer, a mixture-density tabular Transformer architecture specifically designed for TL in RR forecasting across heterogeneous feature sets. The model produces both loan-level point estimates and portfolio-level predictive distributions, thereby supporting a wide range of practical RR forecasting applications. We evaluate the proposed approach in a controlled Monte Carlo simulation that facilitates systematic variation of covariate, conditional, and label shifts, as well as in a real-world transfer setting using the Global Credit Data (GCD) loan dataset as source and a novel bonds dataset as target. Our results show that FT-MDN-Transformer outperforms baseline models when target-domain data are limited, with particularly pronounced gains under covariate and conditional shifts, while label shift remains challenging. We also observe its probabilistic forecasts to closely track empirical recovery distributions, providing richer information than conventional point-prediction metrics alone. Overall, the findings highlight the potential of distribution-aware TL architectures to improve RR forecasting in data-scarce credit portfolios and offer practical insights for risk managers operating under heterogeneous data environments.

Randomized controlled trials often do not represent the populations where decisions are made, and covariate shift across studies can invalidate standard IPD meta-analysis and transport estimators. We propose a placebo-anchored transport framework that treats source-trial outcomes as abundant proxy signals and target-trial placebo outcomes as scarce, high-fidelity gold labels to calibrate baseline risk. A low-complexity (sparse) correction anchors proxy outcome models to the target population, and the anchored models are embedded in a cross-fitted doubly robust learner, yielding a Neyman-orthogonal, target-site doubly robust estimator for patient-level heterogeneous treatment effects when target treated outcomes are available. We distinguish two regimes: in connected targets (with a treated arm), the method yields target-identified effect estimates; in disconnected targets (placebo-only), it reduces to a principled screen--then--transport procedure under explicit working-model transport assumptions. Experiments on synthetic data and a semi-synthetic IHDP benchmark evaluate pointwise CATE accuracy, ATE error, ranking quality for targeting, decision-theoretic policy regret, and calibration. Across connected settings, the proposed method is best or near-best and improves substantially over proxy-only, target-only, and transport baselines at small target sample sizes; in disconnected settings, it retains strong ranking performance for targeting while pointwise accuracy depends on the strength of the working transport condition.

This paper focuses on the problem of unbounded density ratio estimation -- an understudied yet critical challenge in statistical learning -- and its application to covariate shift adaptation. Much of the existing literature assumes that the density ratio is either uniformly bounded or unbounded but known exactly. These conditions are often violated in practice, creating a gap between theoretical guarantees and real-world applicability. In contrast, this work directly addresses unbounded density ratios and integrates them into importance weighting for effective covariate shift adaptation. We propose a three-step estimation method that leverages unlabeled data from both the source and target distributions: (1) estimating a relative density ratio; (2) applying a truncation operation to control its unboundedness; and (3) transforming the truncated estimate back into the standard density ratio. The estimated density ratio is then employed as importance weights for regression under covariate shift. We establish rigorous, non-asymptotic convergence guarantees for both the proposed density ratio estimator and the resulting regression function estimator, demonstrating optimal or near-optimal convergence rates. Our findings offer new theoretical insights into density ratio estimation and learning under covariate shift, extending classical learning theory to more practical and challenging scenarios.

Signal Temporal Logic (STL) inference learns interpretable logical rules for temporal behaviors in dynamical systems. To ensure the correctness of learned STL formulas, recent approaches have incorporated conformal prediction as a statistical tool for uncertainty quantification. However, most existing methods rely on the assumption that calibration and testing data are identically distributed and exchangeable, an assumption that is frequently violated in real-world settings. This paper proposes a conformalized STL inference framework that explicitly addresses covariate shift between training and deployment trajectories dataset. From a technical standpoint, the approach first employs a template-free, differentiable STL inference method to learn an initial model, and subsequently refines it using a limited deployment side dataset to promote distribution alignment. To provide validity guarantees under distribution shift, the framework estimates the likelihood ratio between training and deployment distributions and integrates it into an STL-robustness-based weighted conformal prediction scheme. Experimental results on trajectory datasets demonstrate that the proposed framework preserves the interpretability of STL formulas while significantly improving symbolic learning reliability at deployment time.

Uncertainty quantification is essential for deploying machine learning models in high-stakes domains such as scientific discovery and healthcare. Conformal Prediction (CP) provides finite-sample coverage guarantees under exchangeability, an assumption often violated in practice due to distribution shift. Under covariate shift, restoring validity requires importance weighting, yet accurate density-ratio estimation becomes unstable when training and test distributions exhibit limited support overlap. We propose KMM-CP, a conformal prediction framework based on Kernel Mean Matching (KMM) for covariate-shift correction. We show that KMM directly controls the bias-variance components governing conformal coverage error by minimizing RKHS moment discrepancy under explicit weight constraints, and establish asymptotic coverage guarantees under mild conditions. We then introduce a selective extension that identifies regions of reliable support overlap and restricts conformal correction to this subset, further improving stability in low-overlap regimes. Experiments on molecular property prediction benchmarks with realistic distribution shifts show that KMM-CP reduces coverage gap by over 50% compared to existing approaches. The code is available at https://github.com/siddharthal/KMM-CP.

We propose a principled framework for unsupervised domain adaptation under covariate shift in kernel Generalized Linear Models (GLMs), encompassing kernelized linear, logistic, and Poisson regression with ridge regularization. Our goal is to minimize prediction error in the target domain by leveraging labeled source data and unlabeled target data, despite differences in covariate distributions. We partition the labeled source data into two batches: one for training a family of candidate models, and the other for building an imputation model. This imputation model generates pseudo-labels for the target data, enabling robust model selection. We establish non-asymptotic excess-risk bounds that characterize adaptation performance through an "effective labeled sample size", explicitly accounting for the unknown covariate shift. Experiments on synthetic and real datasets demonstrate consistent performance gains over source-only baselines.

For reliable deployment of deep-learning systems, out-of-distribution (OOD) detection is indispensable. In the real world, where test-time inputs often arrive as streaming mixtures of in-distribution (ID) and OOD samples under evolving covariate shifts, OOD samples are domain-constrained and bounded by the environment, and both ID and OOD are jointly affected by the same covariate factors. Existing methods typically assume a stationary ID distribution, but this assumption breaks down in such settings, leading to severe performance degradation. We empirically discover that, even under covariate shift, covariate-shifted ID (csID) and OOD (csOOD) samples remain separable along a discriminative axis in feature space. Building on this observation, we propose DART, a test-time, online OOD detection method that dynamically tracks dual prototypes -- one for ID and the other for OOD -- to recover the drifting discriminative axis, augmented with multi-layer fusion and flip correction for robustness. Extensive experiments on a wide range of challenging benchmarks, where all datasets are subjected to 15 common corruption types at severity level 5, demonstrate that our method significantly improves performance, yielding 15.32 percentage points (pp) AUROC gain and 49.15 pp FPR@95TPR reduction on ImageNet-C vs. Textures-C compared to established baselines. These results highlight the potential of the test-time discriminative axis tracking for dependable OOD detection in dynamically changing environments.

This work presents Causal Drift Generator (CaDrift), a time-dependent synthetic data generator framework based on Structural Causal Models (SCMs). The framework produces a virtually infinite combination of data streams with controlled shift events and time-dependent data, making it a tool to evaluate methods under evolving data. CaDrift synthesizes various distributional and covariate shifts by drifting mapping functions of the SCM, which change underlying cause-and-effect relationships between features and the target. In addition, CaDrift models occasional perturbations by leveraging interventions in causal modeling. Experimental results show that, after distributional shift events, the accuracy of classifiers tends to drop, followed by a gradual retrieval, confirming the generator's effectiveness in simulating shifts. The framework has been made available on GitHub.

Conformal prediction (CP) offers distribution-free marginal coverage guarantees under an exchangeability assumption, but these guarantees can fail if the data distribution shifts. We analyze the use of pseudo-calibration as a tool to counter this performance loss under a bounded label-conditional covariate shift model. Using tools from domain adaptation, we derive a lower bound on target coverage in terms of the source-domain loss of the classifier and a Wasserstein measure of the shift. Using this result, we provide a method to design pseudo-calibrated sets that inflate the conformal threshold by a slack parameter to keep target coverage above a prescribed level. Finally, we propose a source-tuned pseudo-calibration algorithm that interpolates between hard pseudo-labels and randomized labels as a function of classifier uncertainty. Numerical experiments show that our bounds qualitatively track pseudo-calibration behavior and that the source-tuned scheme mitigates coverage degradation under distribution shift while maintaining nontrivial prediction set sizes.