arXiv subjects

Sivaraman Balakrishnan

Publications and source records attributed to Sivaraman Balakrishnan.

At least 37 records · Page 2Linked to original sources

arXiv2024

Two-Sample Testing with a Graph-Based Total Variation Integral Probability Metric

We consider a novel multivariate nonparametric two-sample testing problem where, under the alternative, distributions $P$ and $Q$ are separated in an integral probability metric over functions of bounded total variation (TV IPM). We propose a new test, the graph TV test, which uses a graph-based approximation to the TV IPM as its test statistic. We show that this test, computed with an $\varepsilon$-neighborhood graph and calibrated by permutation, is minimax rate-optimal for detecting alternatives separated in the TV IPM. As an important special case, we show that this implies the graph TV test is optimal for detecting spatially localized alternatives, whereas the $χ^2$ test is provably suboptimal. Our theory is supported with numerical experiments on simulated and real data.

math.ST↗

arXiv2024

Central Limit Theorems for Smooth Optimal Transport Maps

One of the central objects in the theory of optimal transport is the Brenier map: the unique monotone transformation which pushes forward an absolutely continuous probability law onto any other given law. A line of recent work has analyzed $L^2$ convergence rates of plugin estimators of Brenier maps, which are defined as the Brenier map between density estimators of the underlying distributions. In this work, we show that such estimators satisfy a pointwise central limit theorem when the underlying laws are supported on the flat torus of dimension $d \geq 3$. We also derive a negative result, showing that these estimators do not converge weakly in $L^2$ when the dimension is sufficiently large. Our proofs hinge upon a quantitative linearization of the Monge-Ampère equation, which may be of independent interest. This result allows us to reduce our problem to that of deriving limit laws for the solution of a uniformly elliptic partial differential equation with a stochastic right-hand side, subject to periodic boundary conditions.

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arXiv2024

Causal Effect Estimation after Propensity Score Trimming with Continuous Treatments

Propensity score trimming, which discards subjects with propensity scores below a threshold, is a common way to address positivity violations that complicate causal effect estimation. However, most works on trimming assume treatment is discrete and models for the outcome regression and propensity score are parametric. This work proposes nonparametric estimators for trimmed average causal effects in the case of continuous treatments based on efficient influence functions. For continuous treatments, an efficient influence function for a trimmed causal effect does not exist, due to a lack of pathwise differentiability induced by trimming and a continuous treatment. Thus, we target a smoothed version of the trimmed causal effect for which an efficient influence function exists. Our resulting estimators exhibit doubly-robust style guarantees, with error involving products or squares of errors for the outcome regression and propensity score, which allows for valid inference even when nonparametric models are used. Our results allow the trimming threshold to be fixed or defined as a quantile of the propensity score, such that confidence intervals incorporate uncertainty involved in threshold estimation. These findings are validated via simulation and an application, thereby showing how to efficiently-but-flexibly estimate trimmed causal effects with continuous treatments.

stat.ME↗

arXiv2024

Plugin Estimation of Smooth Optimal Transport Maps

We analyze a number of natural estimators for the optimal transport map between two distributions and show that they are minimax optimal. We adopt the plugin approach: our estimators are simply optimal couplings between measures derived from our observations, appropriately extended so that they define functions on $\mathbb{R}^d$. When the underlying map is assumed to be Lipschitz, we show that computing the optimal coupling between the empirical measures, and extending it using linear smoothers, already gives a minimax optimal estimator. When the underlying map enjoys higher regularity, we show that the optimal coupling between appropriate nonparametric density estimates yields faster rates. Our work also provides new bounds on the risk of corresponding plugin estimators for the quadratic Wasserstein distance, and we show how this problem relates to that of estimating optimal transport maps using stability arguments for smooth and strongly convex Brenier potentials. As an application of our results, we derive central limit theorems for plugin estimators of the squared Wasserstein distance, which are centered at their population counterpart when the underlying distributions have sufficiently smooth densities. In contrast to known central limit theorems for empirical estimators, this result easily lends itself to statistical inference for the quadratic Wasserstein distance.

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arXiv2024

Semi-Supervised U-statistics

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.

math.ST↗

arXiv2023

Minimax rates for heterogeneous causal effect estimation

Estimation of heterogeneous causal effects - i.e., how effects of policies and treatments vary across subjects - is a fundamental task in causal inference. Many methods for estimating conditional average treatment effects (CATEs) have been proposed in recent years, but questions surrounding optimality have remained largely unanswered. In particular, a minimax theory of optimality has yet to be developed, with the minimax rate of convergence and construction of rate-optimal estimators remaining open problems. In this paper we derive the minimax rate for CATE estimation, in a Holder-smooth nonparametric model, and present a new local polynomial estimator, giving high-level conditions under which it is minimax optimal. Our minimax lower bound is derived via a localized version of the method of fuzzy hypotheses, combining lower bound constructions for nonparametric regression and functional estimation. Our proposed estimator can be viewed as a local polynomial R-Learner, based on a localized modification of higher-order influence function methods. The minimax rate we find exhibits several interesting features, including a non-standard elbow phenomenon and an unusual interpolation between nonparametric regression and functional estimation rates. The latter quantifies how the CATE, as an estimand, can be viewed as a regression/functional hybrid.

math.ST↗

arXiv2023

Complementary Benefits of Contrastive Learning and Self-Training Under Distribution Shift

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.

cs.LG↗

arXiv2023

Conditional Independence Testing for Discrete Distributions: Beyond $χ^2$- and $G$-tests

This paper is concerned with the problem of conditional independence testing for discrete data. In recent years, researchers have shed new light on this fundamental problem, emphasizing finite-sample optimality. The non-asymptotic viewpoint adapted in these works has led to novel conditional independence tests that enjoy certain optimality under various regimes. Despite their attractive theoretical properties, the considered tests are not necessarily practical, relying on a Poissonization trick and unspecified constants in their critical values. In this work, we attempt to bridge the gap between theory and practice by reproving optimality without Poissonization and calibrating tests using Monte Carlo permutations. Along the way, we also prove that classical asymptotic $χ^2$- and $G$-tests are notably sub-optimal in a high-dimensional regime, which justifies the demand for new tools. Our theoretical results are complemented by experiments on both simulated and real-world datasets. Accompanying this paper is an R package UCI that implements the proposed tests.

math.ST↗

arXiv2023

Conservative Inference for Counterfactuals

In causal inference, the joint law of a set of counterfactual random variables is generally not identified. We show that a conservative version of the joint law - corresponding to the smallest treatment effect - is identified. Finding this law uses recent results from optimal transport theory. Under this conservative law we can bound causal effects and we may construct inferences for each individual's counterfactual dose-response curve. Intuitively, this is the flattest counterfactual curve for each subject that is consistent with the distribution of the observables. If the outcome is univariate then, under mild conditions, this curve is simply the quantile function of the counterfactual distribution that passes through the observed point. This curve corresponds to a nonparametric rank preserving structural model.

stat.ME↗

arXiv2023

The HulC: Confidence Regions from Convex Hulls

We develop and analyze the HulC, an intuitive and general method for constructing confidence sets using the convex hull of estimates constructed from subsets of the data. Unlike classical methods which are based on estimating the (limiting) distribution of an estimator, the HulC is often simpler to use and effectively bypasses this step. In comparison to the bootstrap, the HulC requires fewer regularity conditions and succeeds in many examples where the bootstrap provably fails. Unlike subsampling, the HulC does not require knowledge of the rate of convergence of the estimators on which it is based. The validity of the HulC requires knowledge of the (asymptotic) median-bias of the estimators. We further analyze a variant of our basic method, called the Adaptive HulC, which is fully data-driven and estimates the median-bias using subsampling. We show that the Adaptive HulC retains the aforementioned strengths of the HulC. In certain cases where the underlying estimators are pathologically asymmetric the HulC and Adaptive HulC can fail to provide useful confidence sets. We propose a final variant, the Unimodal HulC, which can salvage the situation in cases where the distribution of the underlying estimator is (asymptotically) unimodal. We discuss these methods in the context of several challenging inferential problems which arise in parametric, semi-parametric, and non-parametric inference. Although our focus is on validity under weak regularity conditions, we also provide some general results on the width of the HulC confidence sets, showing that in many cases the HulC confidence sets have near-optimal width.

math.ST↗

arXiv2023

Nearly Minimax Optimal Wasserstein Conditional Independence Testing

This paper is concerned with minimax conditional independence testing. In contrast to some previous works on the topic, which use the total variation distance to separate the null from the alternative, here we use the Wasserstein distance. In addition, we impose Wasserstein smoothness conditions which on bounded domains are weaker than the corresponding total variation smoothness imposed, for instance, by Neykov et al. [2021]. This added flexibility expands the distributions which are allowed under the null and the alternative to include distributions which may contain point masses for instance. We characterize the optimal rate of the critical radius of testing up to logarithmic factors. Our test statistic which nearly achieves the optimal critical radius is novel, and can be thought of as a weighted multi-resolution version of the U-statistic studied by Neykov et al. [2021].

math.ST↗

arXiv2023

RLSbench: Domain Adaptation Under Relaxed Label Shift

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.

cs.LG↗

arXiv2023

Online Label Shift: Optimal Dynamic Regret meets Practical Algorithms

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.

stat.ML↗

arXiv2023

Domain Adaptation under Missingness Shift

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.

cs.LG↗

arXiv2022

Gaussian Universal Likelihood Ratio Testing

The classical likelihood ratio test (LRT) based on the asymptotic chi-squared distribution of the log likelihood is one of the fundamental tools of statistical inference. A recent universal LRT approach based on sample splitting provides valid hypothesis tests and confidence sets in any setting for which we can compute the split likelihood ratio statistic (or, more generally, an upper bound on the null maximum likelihood). The universal LRT is valid in finite samples and without regularity conditions. This test empowers statisticians to construct tests in settings for which no valid hypothesis test previously existed. For the simple but fundamental case of testing the population mean of d-dimensional Gaussian data with identity covariance matrix, the classical LRT itself applies. Thus, this setting serves as a perfect test bed to compare the classical LRT against the universal LRT. This work presents the first in-depth exploration of the size, power, and relationships between several universal LRT variants. We show that a repeated subsampling approach is the best choice in terms of size and power. For large numbers of subsamples, the repeated subsampling set is approximately spherical. We observe reasonable performance even in a high-dimensional setting, where the expected squared radius of the best universal LRT's confidence set is approximately 3/2 times the squared radius of the classical LRT's spherical confidence set. We illustrate the benefits of the universal LRT through testing a non-convex doughnut-shaped null hypothesis, where a universal inference procedure can have higher power than a standard approach.

stat.ME↗

arXiv2022

Universal Inference

We propose a general method for constructing hypothesis tests and confidence sets that have finite sample guarantees without regularity conditions. We refer to such procedures as "universal." The method is very simple and is based on a modified version of the usual likelihood ratio statistic, that we call "the split likelihood ratio test" (split LRT). The method is especially appealing for irregular statistical models. Canonical examples include mixture models and models that arise in shape-constrained inference. Constructing tests and confidence sets for such models is notoriously difficult. Typical inference methods, like the likelihood ratio test, are not useful in these cases because they have intractable limiting distributions. In contrast, the method we suggest works for any parametric model and also for some nonparametric models. The split LRT can also be used with profile likelihoods to deal with nuisance parameters, and it can also be run sequentially to yield anytime-valid $p$-values and confidence sequences.

math.ST↗

arXiv2022

Domain Adaptation under Open Set Label Shift

We introduce the problem of domain adaptation under Open Set Label Shift (OSLS) where the label distribution can change arbitrarily and a new class may arrive during deployment, but the class-conditional distributions p(x|y) are domain-invariant. OSLS subsumes domain adaptation under label shift and Positive-Unlabeled (PU) learning. The learner's goals here are two-fold: (a) estimate the target label distribution, including the novel class; and (b) learn a target classifier. First, we establish necessary and sufficient conditions for identifying these quantities. Second, motivated by advances in label shift and PU learning, we propose practical methods for both tasks that leverage black-box predictors. Unlike typical Open Set Domain Adaptation (OSDA) problems, which tend to be ill-posed and amenable only to heuristics, OSLS offers a well-posed problem amenable to more principled machinery. Experiments across numerous semi-synthetic benchmarks on vision, language, and medical datasets demonstrate that our methods consistently outperform OSDA baselines, achieving 10--25% improvements in target domain accuracy. Finally, we analyze the proposed methods, establishing finite-sample convergence to the true label marginal and convergence to optimal classifier for linear models in a Gaussian setup. Code is available at https://github.com/acmi-lab/Open-Set-Label-Shift.

cs.LG↗

arXiv2022

Leveraging Unlabeled Data to Predict Out-of-Distribution Performance

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 (Wilds, ImageNet, Breeds, CIFAR, and MNIST). In our experiments, ATC estimates target performance $2$-$4\times$ 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. Code is available at https://github.com/saurabhgarg1996/ATC_code/.

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