Sivaraman Balakrishnan

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.

Recent methodological research in causal inference has focused on effects of stochastic interventions, which assign treatment randomly, often according to subject-specific covariates. In this work, we demonstrate that the usual notion of stochastic interventions have a surprising property: when there is unmeasured confounding, bounds on their effects do not collapse when the policy approaches the observational regime. As an alternative, we propose to study generalized policies, treatment rules that can depend on covariates, the natural value of treatment, and auxiliary randomness. We show that certain generalized policy formulations can resolve the "non-collapsing" bound issue: bounds narrow to a point when the target treatment distribution approaches that in the observed data. Moreover, drawing connections to the theory of optimal transport, we characterize generalized policies that minimize worst-case bound width in various sensitivity analysis models, as well as corresponding sharp bounds on their causal effects. These optimal policies are new, and can have a more parsimonious interpretation compared to their usual stochastic policy analogues. Finally, we develop flexible, efficient, and robust estimators for the sharp nonparametric bounds that emerge from the framework.

We introduce a new notion of regularity of an estimator called median regularity. We prove that uniformly valid honest inference for a functional is possible if and only if there exists a median regular estimator of that functional. To the best of our knowledge, such a notion of regularity that is necessary for uniformly valid inference is unavailable in the literature.

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.

Summary: The classical likelihood ratio test based on the asymptotic chi-squared distribution of the log-likelihood is one of the fundamental tools of statistical inference. A recent universal likelihood ratio test 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 likelihood ratio test 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 an identity covariance matrix, the classical likelihood ratio test itself applies. Thus, this setting serves as a perfect test bed to compare the classical likelihood ratio test against the universal likelihood ratio test. This work presents the first in-depth exploration of the size, power and relationships between several universal likelihood ratio test 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 likelihood ratio test’s confidence set is approximately 3/2 times the squared radius of the classical likelihood ratio test’s spherical confidence set. We illustrate the benefits of the universal likelihood ratio test through testing a nonconvex doughnut-shaped null hypothesis, where a universal inference procedure can have higher power than a standard approach.

Motivated by the growing popularity of variants of the Wasserstein distance in statistics and machine learning, we study statistical inference for the Sliced Wasserstein distance--an easily computable variant of the Wasserstein distance. Specifically, we construct confidence intervals for the Sliced Wasserstein distance which have finite-sample validity under no assumptions or under mild moment assumptions. These intervals are adaptive in length to the regularity of the underlying distributions. We also bound the minimax risk of estimating the Sliced Wasserstein distance, and as a consequence establish that the lengths of our proposed confidence intervals are minimax optimal over appropriate distribution classes. To motivate the choice of these classes, we also study minimax rates of estimating a distribution under the Sliced Wasserstein distance. These theoretical findings are complemented with a simulation study demonstrating the deficiencies of the classical bootstrap, and the advantages of our proposed methods. We also show strong correspondences between our theoretical predictions and the adaptivity of our confidence interval lengths in simulations. We conclude by demonstrating the use of our confidence intervals in the setting of simulator-based likelihood-free inference. In this setting, contrasting popular approximate Bayesian computation methods, we develop uncertainty quantification methods with rigorous frequentist coverage guarantees.

The task of aggregating and denoising crowd-labeled data has gained increased significance with the advent of crowdsourcing platforms and massive datasets. We propose a permutation-based model for crowd labeled data that is a significant generalization of the classical Dawid-Skene model, and introduce a new error metric by which to compare different estimators. We derive global minimax rates for the permutation-based model that are sharp up to logarithmic factors, and match the minimax lower bounds derived under the simpler Dawid-Skene model. We then design two computationally-efficient estimators: the WAN estimator for the setting where the ordering of workers in terms of their abilities is approximately known, and the OBI-WAN estimator where that is not known. For each of these estimators, we provide non-asymptotic bounds on their performance. We conduct synthetic simulations and experiments on real-world crowdsourcing data, and the experimental results corroborate our theoretical findings.

In supervised learning, we typically leverage a fully labeled dataset to design methods for function estimation or prediction. In many practical situations, we are able to obtain alternative feedback, possibly at a low cost. A broad goal is to understand the usefulness of, and to design algorithms to exploit, this alternative feedback. In this paper, we consider a semi-supervised regression setting, where we obtain additional ordinal (or comparison) information for the unlabeled samples. We consider ordinal feedback of varying qualities where we have either a perfect ordering of the samples, a noisy ordering of the samples or noisy pairwise comparisons between the samples. We provide a precise quantification of the usefulness of these types of ordinal feedback in both nonparametric and linear regression, showing that in many cases it is possible to accurately estimate an underlying function with a very small labeled set, effectively \emph{escaping the curse of dimensionality}. We also present lower bounds, that establish fundamental limits for the task and show that our algorithms are optimal in a variety of settings. Finally, we present extensive experiments on new datasets that demonstrate the efficacy and practicality of our algorithms and investigate their robustness to various sources of noise and model misspecification.

We consider the problem of global optimization of an unknown non-convex smooth function with noisy zeroth-order feedback. We propose a local minimax framework to study the fundamental difficulty of optimizing smooth functions with adaptive function evaluations. We show that for functions with fast growth around their global minima, carefully designed optimization algorithms can identify a near global minimizer with many fewer queries than worst-case global minimax theory predicts. For the special case of strongly convex and smooth functions, our implied convergence rates match the ones developed for zeroth-order convex optimization problems. On the other hand, we show that in the worst case no algorithm can converge faster than the minimax rate of estimating an unknown function in the ℓ _∞ -norm. Finally, we show that non-adaptive algorithms, though optimal in a global minimax sense, do not attain the optimal local minimax rate.