We study estimators of the optimal transport (OT) map between two probability distributions. We focus on plugin estimators derived from the OT map between estimates of the underlying distributions. We develop novel stability bounds for OT maps which generalize those in past work, and allow us to reduce the problem of optimally estimating the transport map to that of optimally estimating densities in the Wasserstein distance. In contrast, past work provided a partial connection between these problems and relied on regularity theory for the Monge-Ampere equation to bridge the gap, a step which required unnatural assumptions to obtain sharp guarantees. We also provide some new insights into the connections between stability bounds which arise in the analysis of plugin estimators and growth bounds for the semi-dual functional which arise in the analysis of Brenier potential-based estimators of the transport map. We illustrate the applicability of our new stability bounds by revisiting the smooth setting studied by Manole et al., analyzing two of their estimators under more general conditions. Critically, our bounds do not require smoothness or boundedness assumptions on the underlying measures. As an illustrative application, we develop and analyze a novel tuning parameter-free estimator for the OT map between two strongly log-concave distributions.

Doubly robust estimators with cross-fitting have gained popularity in causal inference due to their favorable structure-agnostic error guarantees. However, when additional structure, such as H\"{o}lder smoothness, is available then more accurate "double cross-fit doubly robust" (DCDR) estimators can be constructed by splitting the training data and undersmoothing nuisance function estimators on independent samples. We study a DCDR estimator of the Expected Conditional Covariance, a functional of interest in causal inference and conditional independence testing, and derive a series of increasingly powerful results with progressively stronger assumptions. We first provide a structure-agnostic error analysis for the DCDR estimator with no assumptions on the nuisance functions or their estimators. Then, assuming the nuisance functions are H\"{o}lder smooth, but without assuming knowledge of the true smoothness level or the covariate density, we establish that DCDR estimators with several linear smoothers are semiparametric efficient under minimal conditions and achieve fast convergence rates in the non-$\sqrt{n}$ regime. When the covariate density and smoothnesses are known, we propose a minimax rate-optimal DCDR estimator based on undersmoothed kernel regression. Moreover, we show an undersmoothed DCDR estimator satisfies a slower-than-$\sqrt{n}$ central limit theorem, and that inference is possible even in the non-$\sqrt{n}$ regime. Finally, we support our theoretical results with simulations, providing intuition for double cross-fitting and undersmoothing, demonstrating where our estimator achieves semiparametric efficiency while the usual "single cross-fit" estimator fails, and illustrating asymptotic normality for the undersmoothed DCDR estimator.

Semi-supervised datasets are ubiquitous across diverse domains where obtaining fully labeled data is costly or time-consuming. The prevalence of such datasets has consistently driven the demand for new tools and methods that exploit the potential of unlabeled data. Responding to this demand, we introduce semi-supervised U-statistics enhanced by the abundance of unlabeled data, and investigate their statistical properties. We show that the proposed approach is asymptotically Normal and exhibits notable efficiency gains over classical U-statistics by effectively integrating various powerful prediction tools into the framework. To understand the fundamental difficulty of the problem, we derive minimax lower bounds in semi-supervised settings and showcase that our procedure is semi-parametrically efficient under regularity conditions. Moreover, tailored to bivariate kernels, we propose a refined approach that outperforms the classical U-statistic across all degeneracy regimes, and demonstrate its optimality properties. Simulation studies are conducted to corroborate our findings and to further demonstrate our framework.

Self-training and contrastive learning have emerged as leading techniques for incorporating unlabeled data, both under distribution shift (unsupervised domain adaptation) and when it is absent (semi-supervised learning). However, despite the popularity and compatibility of these techniques, their efficacy in combination remains unexplored. In this paper, we undertake a systematic empirical investigation of this combination, finding that (i) in domain adaptation settings, self-training and contrastive learning offer significant complementary gains; and (ii) in semi-supervised learning settings, surprisingly, the benefits are not synergistic. Across eight distribution shift datasets (e.g., BREEDs, WILDS), we demonstrate that the combined method obtains 3--8% higher accuracy than either approach independently. We then theoretically analyze these techniques in a simplified model of distribution shift, demonstrating scenarios under which the features produced by contrastive learning can yield a good initialization for self-training to further amplify gains and achieve optimal performance, even when either method alone would fail.

Despite the emergence of principled methods for domain adaptation under label shift, their sensitivity to shifts in class conditional distributions is precariously under explored. Meanwhile, popular deep domain adaptation heuristics tend to falter when faced with label proportions shifts. While several papers modify these heuristics in attempts to handle label proportions shifts, inconsistencies in evaluation standards, datasets, and baselines make it difficult to gauge the current best practices. In this paper, we introduce RLSbench, a large-scale benchmark for relaxed label shift, consisting of $>$500 distribution shift pairs spanning vision, tabular, and language modalities, with varying label proportions. Unlike existing benchmarks, which primarily focus on shifts in class-conditional $p(x|y)$, our benchmark also focuses on label marginal shifts. First, we assess 13 popular domain adaptation methods, demonstrating more widespread failures under label proportion shifts than were previously known. Next, we develop an effective two-step meta-algorithm that is compatible with most domain adaptation heuristics: (i) pseudo-balance the data at each epoch; and (ii) adjust the final classifier with target label distribution estimate. The meta-algorithm improves existing domain adaptation heuristics under large label proportion shifts, often by 2--10\% accuracy points, while conferring minimal effect ($<$0.5\%) when label proportions do not shift. We hope that these findings and the availability of RLSbench will encourage researchers to rigorously evaluate proposed methods in relaxed label shift settings. Code is publicly available at https://github.com/acmi-lab/RLSbench.

This paper focuses on supervised and unsupervised online label shift, where the class marginals $Q(y)$ varies but the class-conditionals $Q(x|y)$ remain invariant. In the unsupervised setting, our goal is to adapt a learner, trained on some offline labeled data, to changing label distributions given unlabeled online data. In the supervised setting, we must both learn a classifier and adapt to the dynamically evolving class marginals given only labeled online data. We develop novel algorithms that reduce the adaptation problem to online regression and guarantee optimal dynamic regret without any prior knowledge of the extent of drift in the label distribution. Our solution is based on bootstrapping the estimates of \emph{online regression oracles} that track the drifting proportions. Experiments across numerous simulated and real-world online label shift scenarios demonstrate the superior performance of our proposed approaches, often achieving 1-3\% improvement in accuracy while being sample and computationally efficient. Code is publicly available at https://github.com/acmi-lab/OnlineLabelShift.

Many recent developments in causal inference, and functional estimation problems more generally, have been motivated by the fact that classical one-step (first-order) debiasing methods, or their more recent sample-split double machine-learning avatars, can outperform plugin estimators under surprisingly weak conditions. These first-order corrections improve on plugin estimators in a black-box fashion, and consequently are often used in conjunction with powerful off-the-shelf estimation methods. These first-order methods are however provably suboptimal in a minimax sense for functional estimation when the nuisance functions live in Holder-type function spaces. This suboptimality of first-order debiasing has motivated the development of "higher-order" debiasing methods. The resulting estimators are, in some cases, provably optimal over Holder-type spaces, but both the estimators which are minimax-optimal and their analyses are crucially tied to properties of the underlying function space. In this paper we investigate the fundamental limits of structure-agnostic functional estimation, where relatively weak conditions are placed on the underlying nuisance functions. We show that there is a strong sense in which existing first-order methods are optimal. We achieve this goal by providing a formalization of the problem of functional estimation with black-box nuisance function estimates, and deriving minimax lower bounds for this problem. Our results highlight some clear tradeoffs in functional estimation -- if we wish to remain agnostic to the underlying nuisance function spaces, impose only high-level rate conditions, and maintain compatibility with black-box nuisance estimators then first-order methods are optimal. When we have an understanding of the structure of the underlying nuisance functions then carefully constructed higher-order estimators can outperform first-order estimators.

Rates of missing data often depend on record-keeping policies and thus may change across times and locations, even when the underlying features are comparatively stable. In this paper, we introduce the problem of Domain Adaptation under Missingness Shift (DAMS). Here, (labeled) source data and (unlabeled) target data would be exchangeable but for different missing data mechanisms. We show that if missing data indicators are available, DAMS reduces to covariate shift. Addressing cases where such indicators are absent, we establish the following theoretical results for underreporting completely at random: (i) covariate shift is violated (adaptation is required); (ii) the optimal linear source predictor can perform arbitrarily worse on the target domain than always predicting the mean; (iii) the optimal target predictor can be identified, even when the missingness rates themselves are not; and (iv) for linear models, a simple analytic adjustment yields consistent estimates of the optimal target parameters. In experiments on synthetic and semi-synthetic data, we demonstrate the promise of our methods when assumptions hold. Finally, we discuss a rich family of future extensions.

Real-world machine learning deployments are characterized by mismatches between the source (training) and target (test) distributions that may cause performance drops. In this work, we investigate methods for predicting the target domain accuracy using only labeled source data and unlabeled target data. We propose Average Thresholded Confidence (ATC), a practical method that learns a threshold on the model's confidence, predicting accuracy as the fraction of unlabeled examples for which model confidence exceeds that threshold. ATC outperforms previous methods across several model architectures, types of distribution shifts (e.g., due to synthetic corruptions, dataset reproduction, or novel subpopulations), and datasets (W In our experiments, ATC estimates target performance 2-4ˆ more accurately than prior methods. We also explore the theoretical foundations of the problem, proving that, in general, identifying the accuracy is just as hard as identifying the optimal predictor and thus, the efficacy of any method rests upon (perhaps unstated) assumptions on the nature of the shift. Finally, analyzing our method on some toy distributions, we provide insights concerning when it works. Machine learning models deployed in the real world typically encounter examples from previously unseen distributions. While the IID assumption enables us to evaluate models using held-out data from the source distribution (from which training data is sampled), this estimate is no longer valid in presence of a distribution shift. Moreover, under such shifts, model accuracy tends to degrade (Szegedy et al., 2014; Recht et al., 2019; Koh et al., 2021). Commonly, the only data available to the practitioner are a labeled training set (source) and unlabeled deployment-time data which makes the problem more difficult. In this setting, detecting shifts in the distribution of covariates is known to be possible (but difficult) in theory (Ramdas et al., 2015), and in practice (Rabanser et al., 2018). However, producing an optimal predictor using only labeled source and unlabeled target data is well-known to be impossible absent further assumptions (Ben-David et al., 2010; Lipton et al., 2018). Two vital questions that remain are: (i) the precise conditions under which we can estimate a classifier's target-domain accuracy; and (ii) which methods are most practically useful. To begin, the straightforward way to assess the performance of a model under distribution shift would be to collect labeled (target domain) examples and then to evaluate the model on that data. However, collecting fresh labeled data from the target distribution is prohibitively expensive and time-consuming, especially if the target distribution is non-stationary.

In this paper we study the statistical properties of Principal Components Regression with Laplacian Eigenmaps (PCR-LE), a method for nonparametric regression based on Laplacian Eigenmaps (LE). PCR-LE works by projecting a vector of observed responses ${\bf Y} = (Y_1,\ldots,Y_n)$ onto a subspace spanned by certain eigenvectors of a neighborhood graph Laplacian. We show that PCR-LE achieves minimax rates of convergence for random design regression over Sobolev spaces. Under sufficient smoothness conditions on the design density $p$, PCR-LE achieves the optimal rates for both estimation (where the optimal rate in squared $L^2$ norm is known to be $n^{-2s/(2s + d)}$) and goodness-of-fit testing ($n^{-4s/(4s + d)}$). We also show that PCR-LE is \emph{manifold adaptive}: that is, we consider the situation where the design is supported on a manifold of small intrinsic dimension $m$, and give upper bounds establishing that PCR-LE achieves the faster minimax estimation ($n^{-2s/(2s + m)}$) and testing ($n^{-4s/(4s + m)}$) rates of convergence. Interestingly, these rates are almost always much faster than the known rates of convergence of graph Laplacian eigenvectors to their population-level limits; in other words, for this problem regression with estimated features appears to be much easier, statistically speaking, than estimating the features itself. We support these theoretical results with empirical evidence.