Average dose-response functions are widely used to summarize causal effects of continuous treatments, but most existing methods assume that the observed sample represents the target population. We study a covariate-shift setting in which covariates, treatment, and outcome are observed in a labelled source sample, while only covariates are observed in the target sample. We develop a two-sample local polynomial regression framework based on pseudo-outcomes that use source outcomes to address confounding and target covariates to define the population of interest. We further propose a source-to-target extension of distance covariance optimal weighting (DCOW), designed to remove treatment-covariate dependence in the source sample while aligning the weighted source covariate distribution with the target population. A central theoretical contribution is a weight-level analysis of this optimization-based procedure: we show that the population criterion identifies the oracle source-to-target weights and that approximate empirical minimizers, including exact minimizers as a special case, converge uniformly to these weights under regularity conditions. We also establish consistency and asymptotic normality of the resulting estimator. Simulations show that the proposed method improves target dose-response estimation relative to DCOW, generalized-propensity-score weighting, entropy balancing, and unweighted alternatives. We illustrate the method in a county-level analysis of PM2.5 exposure and subsequent heart-disease mortality using a source-target validation design.
In transfer-learning settings, a model derived from abundant surrogate labels may be deployed in a target population where gold-standard outcomes are unobserved. Evaluating its target performance is essential for determining whether decisions based on the model remain reliable, yet it is difficult when gold labels are scarce, and covariate distributions differ across data sources. We study a three-sample setting with a small gold-labeled source, a larger surrogate-labeled source, and an unlabeled target. Under conditional transportability, we evaluate the surrogate-derived model against the latent gold-standard outcome in the target population. We propose cross-fitted estimators that transport information from the two labeled sources through source-specific density ratios. We also combine outcome-regression augmentation with a kernel correction for estimating the model near a threshold, accounting for uncertainty from all three samples. We establish asymptotically linear inference for TPR and FPR, consistency and pointwise inference for the ROC curve, and asymptotically normal inference for AUC. Simulations assess bias, coverage, and sensitivity to bandwidth and relative sample sizes. A retrospective temporal validation on Chatbot Arena and a semi-synthetic ACS-Income study provide validation in real-world AI applications.
Certified selective predictors attain whatever coverage they attain; operators impose an automation floor: answer at least a $β$-fraction of shifted target traffic with at most an $α$-fraction of answers wrong. Under bounded-ratio covariate shift we prove the Floor Certification Map: once that floor must be certified alongside the selection-conditioned risk $α$, certification acquires a feasibility frontier and a two-resource complexity map, additive up to constants: risk in labeled source, the floor in unlabeled target samples. The rates are local, needing a regular frontier margin, slack below the local-regime threshold, and lattice conditions: pre-registered with a lattice margin for the upper bounds, compatible per-slack for the lower. The displayed split is the operational route; oracle weights also allow a labeled-source floor estimate. Three model-tagged results: a lower bound (Model-B), a matching oracle-weight upper bound (Model-A), and an implementable upper bound (Model-B') valid under a pre-registered exact stratified-shift model with nuisance cost priced explicitly. The match is across these models rather than a single-model minimax theorem, and necessarily so: over the full bounded-ratio class no unknown-weight procedure matches at any sample size (Model-B is inconsistent, witnessed at $α=β=1/2$). The nuisance's necessity is only partially settled. Complexity tracks a localized accepted-region functional, not global effective sample size (ESS), on both sides, though a fixed-ESS separation theorem is left open; both lower-bound axes vanish as $β\to0$, so the floor creates the map. Empirically, the registered bite family diverges with log-log slope $-2.002$ within its pre-registered band; a 1,024-cell audit records 0 violations where the formal certificates fire; and a single-corpus SQuAD-to-NewsQA feasibility audit returns honest refusal.
This paper develops procedures for nonparametric goodness-of-fit testing under covariate shift, where labelled data are drawn from a source population but goodness-of-fit is evaluated for a target population. The distribution mismatch is quantified by either a bounded moment condition or a sub-exponential tail condition on the target-to-source density ratio. Our method combines truncated importance-weighting kernel ridge regression with a multiplier bootstrap to construct confidence sets for the regression function. The truncation stabilizes the importance- weighting kernel ridge regression as well as the bootstrap calibration, making our approach applicable even when the density ratio has heavy tails. We prove nonasymptotic validity and sharpness of the resulting confidence sets under suitable operator compatibility conditions, and establish explicit error rates for coverage probability under specific conditions on the target- to-source density ratio and on the spectral decay of the kernel integral operator. Numerical experiments corroborate our theoretical findings.
Behraj Khan, Shabir Ahmad, Syed Ahmad Chan Bukhari +1cs.LG
Medical world models aim to learn a latent state of patient or organ physiology and a transition function that forecasts how that state evolves under interventions, supporting downstream tasks from imaging-based diagnosis to digital-twin treatment planning. Two failure modes threaten the reliability of such models in clinical deployment: (i)~\emph{covariate shift}, because training data are fragmented across hospitals, scanners, and time, so the feature distribution seen by the latent-dynamics predictor differs across fragments and from the distribution at deployment; and (ii)~\emph{confidence misalignment}, because multi-step forecasts are often overconfident exactly where clinical risk is highest. We argue that both problems admit a unified treatment via a single lightweight regularisation objective, \textbf{CalTwin}, which combines a Fisher-Information-based shift penalty adapted from our prior work on fragmented covariate-shift remediation~\cite{khan2025mitigating,khan2025causal} with a Confidence Misalignment Penalty adapted from our prior work on calibrated vision-language classification~\cite{khan2025confidence}, applied here to a GRU-based medical world model's latent transition predictor. We derive the combined objective, establish which proof steps transfer from the classification setting without modification and which require adaptation, and evaluate it on the PhysioNet 2019 Sepsis Challenge, treating the two hospital systems as sequential training fragments and the unseen system as an out-of-distribution test. CalTwin reduces OOD next-step latent-state MSE by 9.1\% relative to the no-penalty baseline (FIM penalty alone accounts for 7.0\%); the ECE reduction from the Confidence Misalignment Penalty is real but small (0.7\% for CalTwin, 1.3\% for CMP alone).
Behraj Khan, Behroz Mirza, Syed Ahmad Chan Bukhari +1cs.LG
Covariate shift across training-data partitions biases model selection and parameter estimation in cross-validation, lifelong learning, and federated learning. We propose \textit{Partition-Induced Covariate-shift Correction} (\texttt{PIcsC}), a Fisher information-based regularization framework that mitigates distribution mismatch between data partitions and a reference distribution. \texttt{PIcsC} approximates partition divergence using the Fisher Information Matrix (FIM) and incorporates the resulting statistic as a regularizer during optimization. The same formulation applies to both centrally partitioned datasets (batches or cross-validation folds) and inherently distributed data (federated clients or decentralized nodes), requiring only partition-local gradient statistics rather than raw data. We further introduce a conditional adaptation mechanism that combines FIM shift with KL divergence to detect significant distribution shifts and activates regularization only when necessary. Experiments on more than 40 datasets demonstrate consistent improvements under both natural and synthetic covariate shift. On fragmented batch and fold settings, \texttt{PIcsC} reduces fragmentation-induced performance degradation by more than 20\% and 25\%, respectively. On seven federated learning benchmarks, it consistently outperforms FedAvg, FedProx, and SCAFFOLD by 3 -5 percentage points without requiring client-specific personalization. These results demonstrate that Fisher information provides an effective and unified mechanism for mitigating partition-induced covariate shift across both centralized and distributed learning.
Many covariate-shift adaptation methods construct a correction $w(x)$, but users must still determine whether the corrected distribution is sufficiently balanced for the target stream. We study anytime-valid confirmation of prespecified corrections from sequential unlabeled target inputs. Our primary contribution is a procedure for confirming covariate balance. For a prespecified class of balancing functions and tolerances, time-uniform confidence sequences permit continuous monitoring and data-dependent stopping once all plausible target moments lie within their tolerance bands. If the correction is out of tolerance for at least one function, the probability of ever incorrectly confirming balance is at most the prescribed level. Upon stopping, the procedure yields a certificate local to the chosen functions and tolerances, yet providing an absolute downstream-adequacy statement that ordinary shift diagnostics generally do not. With finite source data, contracted bands preserve this guarantee while accounting for uncertainty in weighted source moments, whereas expanded bands support only compatibility diagnostics. As complementary information, we study a source-calibrated likelihood-ratio e-process whose KL-drift identity characterizes correction directions relative to the source. Under the source-reference distribution, the probability of ever crossing its evidence threshold is controlled, but crossing does not confirm balance. We also give an exponential-tilt test for departures beyond an acceptable correction region and deploy balance-confirmed corrections in weighted conformal prediction. Experiments illustrate false-confirmation control, locality to the balancing-function class, KL-drift diagnostics, acceptable-region monitoring, finite-source effects, and downstream conformal coverage under covariate shift.
Covariate shift often occurs because, in many real applications, the source and the target observations may be generated from different distributions. In this case, the standard metric under the source distribution is not appropriate. This paper considers deep neural network estimators for nonparametric quantile and Huber regression under covariate shift and from dependent observations. We deal with a generalized Bernstein-type inequality that is satisfied by many classical models, including i.i.d. observations, $φ$-mixing, strong mixing, and $\mathcal{C}$-mixing processes. To perform the covariate shift phenomenon, we propose a sparse-penalized deep neural network (SPDNN) estimator that takes into account the discrepancy between the source and target distributions of the data. When the density ratio (between the source and target distributions of the covariate) is unknown, a two steps pre-training procedure is carried out: the first step is devoted to the construction of a least squares SPDNN estimator of the density ratio; which is used in the second step to perform a pre-training reweighted SPDNN estimator of the regression function. For both the quantile and the Huber regression, non-asymptotic error bounds of the proposed SPDNN estimators are established in the class of Hölder smooth functions. These estimators can adaptively attain (up to a logarithmic factor) the minimax optimal convergence rate from i.i.d. data as well as from several classical time series models.
Reliable confidence estimates are essential in semantic segmentation, especially in safety-critical settings where overconfident errors can mislead downstream decisions. Yet modern segmentation models often remain miscalibrated. Post-hoc calibration offers a practical way to correct confidence estimates without retraining the segmentation model, but its use in dense prediction raises structural issues that are often overlooked. We study two such issues. First, adding a constant to all logits leaves the softmax probabilities unchanged, but several standard calibrators can still depend on this arbitrary offset. As a result, two logit representations encoding the same predictive distribution may yield different calibrated probabilities. We define translation-invariant (TI) calibrators as those whose outputs are unchanged under such shifts, characterize which common calibrators satisfy this property, and construct TI counterparts of shift-sensitive calibrators to isolate the effect of removing representation dependence. Second, post-hoc calibration is typically fitted by minimizing a likelihood-based objective, whereas segmentation models are trained with task-specific metrics such as Dice. This mismatch can cause calibration to alter class orderings and degrade the deployed segmentation map. We study decision-preserving calibration under argmax- and order-preservation constraints. Since enforcing these constraints collapses affine softmax calibrators to temperature scaling, we introduce class-conditional affine calibrators that can be made argmax- or order-preserving while retaining greater expressivity, allowing us to quantify the calibration-segmentation trade-off induced by decision preservation. Across natural-image and medical segmentation benchmarks, and under corruption-based covariate shift, matched comparisons show that TI variants generally improve calibration metrics, while decision-preserving variants prevent segmentation degradation and retain strong calibration performance. These results provide practical design principles for well-defined post-hoc calibration pipelines in semantic segmentation.
Distribution shift between training and deployment is a pervasive challenge for modern AI systems. In many cases, the target marginals of covariates and response are known or specified through population-level observations, boundary conditions, properties of simulator configurations, or alignment-time distributional constraints. Such knowledge may provide valuable side information for regression estimation. We study this problem in the multivariate linear regression setting with a stable conditional mean $E[Y\mid X]$ across source and target, and identify the hybrid-loss estimator, which jointly incorporates both target marginals, as a benchmark target-aware estimator. Its direct computation, however, requires solving a coupled nonlinear optimization that is expensive at scale. Our main contribution is to develop and evaluate two computationally tractable alternatives: a constrained moment-matching estimator and a two-stage estimator that augments ordinary least squares with a calibration step. For all three estimators, we derive and compare closed-form asymptotic mean squared errors, yielding conditions under which the tractable alternatives match or closely approximate the hybrid benchmark, and regimes in which they do not. Monte Carlo experiments across three controlled shift regimes validate the theoretical results, investigate the accuracy-runtime tradeoffs among the three estimators, and translate into guidance on estimator choice. In particular, the two-stage estimator nearly matches the hybrid benchmark in the high signal-to-noise regime at essentially no additional cost, providing theoretical grounding for empirical observations in nonlinear settings.
Hanna Myleiko, Sergei Solodky, Vasyl Semenovstat.ML cs.LG math.NA
This paper investigates convergence properties of regularized Nyström subsampling applied to the unsupervised domain adaptation problem under covariate shift. We focus on the low-smoothness (misspecified) case where the target function lies outside the reproducing kernel Hilbert space. By combining Tikhonov regularization with Nyström projection onto a subsampled subspace, we obtain upper bounds on the excess risk that hold with high probability and are expressed in terms of the source condition, the effective dimension, and the sample sizes. We further extend the analysis to the setting where the Radon-Nikodym derivative between the target and source marginal distributions is unknown and must be approximated, and we identify the minimal additional sample sizes required to maintain the same convergence rate as in the oracle case.
Offline model-based optimization (MBO) proposes candidates by optimizing a surrogate trained on a fixed historical dataset. Because candidates are deliberately out-of-distribution, surrogate rankings are least reliable exactly where the optimizer is most aggressive, yet existing methods provide no per-candidate statistical certificate that a design meets a target threshold. We propose \emph{Conformal Candidate Certification} (CCC), a post-hoc wrapper that attaches a calibrated one-sided lower bound to each candidate and advances only those whose bound exceeds the target. We show that entropy-regularized surrogate maximization induces a Gibbs-tilted proposal, so the same surrogate supplies importance weights for weighted conformal prediction without a separate density-ratio estimation step. In a controlled synthetic study, CCC certifies $16.7\%$ of an aggressive proposal pool with empirical coverage 0.990 at nominal 0.90, while standard conformal prediction ignoring the covariate shift collapses to 0.416 coverage.
Bradley G. Karat, Maëliss Jallais, Ali R. Khan +3cs.LG physics.med-ph
Diffusion MRI enables non-invasive probing of tissue microstructure, but accurate parameter estimation is challenged by noise-related effects. In supervised machine learning frameworks trained on simulated data, discrepancies between the noise characteristics of simulated and acquired signals introduce a form of covariate shift, whereby the input signal distribution differs between training and inference. We investigated the impact of this mismatch on microstructure parameter estimation and propose a realistic noise synthesis (RNS) framework to mitigate it. RNS incorporates both the Rician expectation and the effective post-processing noise variance into simulated training signals. The Rician expectation was modelled using a noise standard deviation estimated with MPPCA, while the effective standard deviation was derived from spherical harmonic residuals of preprocessed data. The method was evaluated using the cylinder-zeppelin and the SANDI models on simulated datasets across multiple SNR levels and on in vivo diffusion data with repeated acquisitions. Sensitivity to noise misestimation was also assessed. Ignoring magnitude-induced noise effects during training produced systematic, SNR-dependent parameter bias, particularly at low SNR. Incorporating the Rician expectation substantially reduced bias to the level of noise-aware nonlinear least-squares fitting. Modelling the effective standard deviation further improved precision. Performance was largely independent of regression architecture but sensitive to accurate noise estimation. These findings demonstrate that realistic noise modelling in simulated training data mitigates signal-domain covariate shift and is essential for unbiased supervised microstructure estimation, particularly in low-SNR regimes associated with high b-values or high spatial resolution.
We study long-horizon deployment of a frozen predictor under dynamic covariate shift. A time-domain Poincaré inequality reduces temporal risk volatility to derivative energy, and a Jacobian-velocity theorem identifies directional tangent energy along the deployment path as the governing quantity under explicit along-path regularity and domination assumptions. 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, beats isotropic smoothing there, and gives validation-selected deployment gains on both real datasets when the Air Quality drift subspace is estimated from target-orthogonal sensor motion. Moderate drift-subspace misspecification is tolerable while orthogonal misspecification largely removes the benefit.