Xiaojie Mao

Andrew Bennett, Nathan Kallus, Xiaojie Mao et al. · harvard

In this paper, we study nonparametric estimation of instrumental variable (IV) regressions. Recently, many flexible machine learning methods have been developed for instrumental variable estimation. However, these methods have at least one of the following limitations: (1) restricting the IV regression to be uniquely identified; (2) only obtaining estimation error rates in terms of pseudometrics (\emph{e.g.,} projected norm) rather than valid metrics (\emph{e.g.,} $L_2$ norm); or (3) imposing the so-called closedness condition that requires a certain conditional expectation operator to be sufficiently smooth. In this paper, we present the first method and analysis that can avoid all three limitations, while still permitting general function approximation. Specifically, we propose a new penalized minimax estimator that can converge to a fixed IV solution even when there are multiple solutions, and we derive a strong $L_2$ error rate for our estimator under lax conditions. Notably, this guarantee only needs a widely-used source condition and realizability assumptions, but not the so-called closedness condition. We argue that the source condition and the closedness condition are inherently conflicting, so relaxing the latter significantly improves upon the existing literature that requires both conditions. Our estimator can achieve this improvement because it builds on a novel formulation of the IV estimation problem as a constrained optimization problem.

Andrew Bennett, Nathan Kallus, Xiaojie Mao et al. · harvard

We consider estimation of parameters defined as linear functionals of solutions to linear inverse problems. Any such parameter admits a doubly robust representation that depends on the solution to a dual linear inverse problem, where the dual solution can be thought as a generalization of the inverse propensity function. We provide the first source condition double robust inference method that ensures asymptotic normality around the parameter of interest as long as either the primal or the dual inverse problem is sufficiently well-posed, without knowledge of which inverse problem is the more well-posed one. Our result is enabled by novel guarantees for iterated Tikhonov regularized adversarial estimators for linear inverse problems, over general hypothesis spaces, which are developments of independent interest.

Jian Chen, Zhehao Li, Xiaojie Mao

We study the problem of classification with selectively labeled data, whose distribution may differ from the full population due to historical decision-making. We exploit the fact that in many applications historical decisions were made by multiple decision-makers, each with different decision rules. We analyze this setup under a principled instrumental variable (IV) framework and rigorously study the identification of classification risk. We establish conditions for the exact identification of classification risk and derive tight partial identification bounds when exact identification fails. We further propose a unified cost-sensitive learning (UCL) approach to learn classifiers robust to selection bias in both identification settings. Finally, we theoretically and numerically validate the efficacy of our proposed method.

Yong Liang, Xiaojie Mao, Shiyuan Wang

We study an online joint assortment-inventory optimization problem, in which we assume that the choice behavior of each customer follows the Multinomial Logit (MNL) choice model, and the attraction parameters are unknown a priori. The retailer makes periodic assortment and inventory decisions to dynamically learn from the customer choice observations about the attraction parameters while maximizing the expected total profit over time. In this paper, we propose a novel algorithm that can effectively balance exploration and exploitation in the online decision-making of assortment and inventory. Our algorithm builds on a new estimator for the MNL attraction parameters, an innovative approach to incentivize exploration by adaptively tuning certain known and unknown parameters, and an optimization oracle to static single-cycle assortment-inventory planning problems with given parameters. We establish a regret upper bound for our algorithm and a lower bound for the online joint assortment-inventory optimization problem, suggesting that our algorithm achieves nearly optimal regret rate, provided that the static optimization oracle is exact. Then we incorporate more practical approximate static optimization oracles into our algorithm, and bound from above the impact of static optimization errors on the regret of our algorithm. We perform numerical studies to demonstrate the effectiveness of our proposed algorithm. At last, we extend our study by incorporating inventory carryover and the learning of customer arrival distribution.

Nathan Kallus, Xiaojie Mao, Kaiwen Wang et al.

Off-policy evaluation and learning (OPE/L) use offline observational data to make better decisions, which is crucial in applications where online experimentation is limited. However, depending entirely on logged data, OPE/L is sensitive to environment distribution shifts -- discrepancies between the data-generating environment and that where policies are deployed. \citet{si2020distributional} proposed distributionally robust OPE/L (DROPE/L) to address this, but the proposal relies on inverse-propensity weighting, whose estimation error and regret will deteriorate if propensities are nonparametrically estimated and whose variance is suboptimal even if not. For standard, non-robust, OPE/L, this is solved by doubly robust (DR) methods, but they do not naturally extend to the more complex DROPE/L, which involves a worst-case expectation. In this paper, we propose the first DR algorithms for DROPE/L with KL-divergence uncertainty sets. For evaluation, we propose Localized Doubly Robust DROPE (LDR$^2$OPE) and show that it achieves semiparametric efficiency under weak product rates conditions. Thanks to a localization technique, LDR$^2$OPE only requires fitting a small number of regressions, just like DR methods for standard OPE. For learning, we propose Continuum Doubly Robust DROPL (CDR$^2$OPL) and show that, under a product rate condition involving a continuum of regressions, it enjoys a fast regret rate of $\mathcal{O}\left(N^{-1/2}\right)$ even when unknown propensities are nonparametrically estimated. We empirically validate our algorithms in simulations and further extend our results to general $f$-divergence uncertainty sets.

Guido Imbens, Nathan Kallus, Xiaojie Mao et al.

We study the identification and estimation of long-term treatment effects when both experimental and observational data are available. Since the long-term outcome is observed only after a long delay, it is not measured in the experimental data, but only recorded in the observational data. However, both types of data include observations of some short-term outcomes. In this paper, we uniquely tackle the challenge of persistent unmeasured confounders, i.e., some unmeasured confounders that can simultaneously affect the treatment, short-term outcomes and the long-term outcome, noting that they invalidate identification strategies in previous literature. To address this challenge, we exploit the sequential structure of multiple short-term outcomes, and develop three novel identification strategies for the average long-term treatment effect. We further propose three corresponding estimators and prove their asymptotic consistency and asymptotic normality. We finally apply our methods to estimate the effect of a job training program on long-term employment using semi-synthetic data. We numerically show that our proposals outperform existing methods that fail to handle persistent confounders.

Nathan Kallus, Xiaojie Mao, Masatoshi Uehara

We study the estimation of causal parameters when not all confounders are observed and instead negative controls are available. Recent work has shown how these can enable identification and efficient estimation via two so-called bridge functions. In this paper, we tackle the primary challenge to causal inference using negative controls: the identification and estimation of these bridge functions. Previous work has relied on completeness conditions on these functions to identify the causal parameters and required uniqueness assumptions in estimation, and they also focused on parametric estimation of bridge functions. Instead, we provide a new identification strategy that avoids the completeness condition. And, we provide new estimators for these functions based on minimax learning formulations. These estimators accommodate general function classes such as Reproducing Kernel Hilbert Spaces and neural networks. We study finite-sample convergence results both for estimating bridge functions themselves and for the final estimation of the causal parameter under a variety of combinations of assumptions. We avoid uniqueness conditions on the bridge functions as much as possible.

Yichun Hu, Nathan Kallus, Xiaojie Mao

Incorporating side observations in decision making can reduce uncertainty and boost performance, but it also requires we tackle a potentially complex predictive relationship. While one may use off-the-shelf machine learning methods to separately learn a predictive model and plug it in, a variety of recent methods instead integrate estimation and optimization by fitting the model to directly optimize downstream decision performance. Surprisingly, in the case of contextual linear optimization, we show that the naive plug-in approach actually achieves regret convergence rates that are significantly faster than methods that directly optimize downstream decision performance. We show this by leveraging the fact that specific problem instances do not have arbitrarily bad near-dual-degeneracy. While there are other pros and cons to consider as we discuss and illustrate numerically, our results highlight a nuanced landscape for the enterprise to integrate estimation and optimization. Our results are overall positive for practice: predictive models are easy and fast to train using existing tools, simple to interpret, and, as we show, lead to decisions that perform very well.

Nathan Kallus, Xiaojie Mao

We study contextual stochastic optimization problems, where we leverage rich auxiliary observations (e.g., product characteristics) to improve decision making with uncertain variables (e.g., demand). We show how to train forest decision policies for this problem by growing trees that choose splits to directly optimize the downstream decision quality, rather than splitting to improve prediction accuracy as in the standard random forest algorithm. We realize this seemingly computationally intractable problem by developing approximate splitting criteria that utilize optimization perturbation analysis to eschew burdensome re-optimization for every candidate split, so that our method scales to large-scale problems. We prove that our splitting criteria consistently approximate the true risk and that our method achieves asymptotic optimality. We extensively validate our method empirically, demonstrating the value of optimization-aware construction of forests and the success of our efficient approximations. We show that our approximate splitting criteria can reduce running time hundredfold, while achieving performance close to forest algorithms that exactly re-optimize for every candidate split.

Nathan Kallus, Xiaojie Mao

In many experimental and observational studies, the outcome of interest is often difficult or expensive to observe, reducing effective sample sizes for estimating average treatment effects (ATEs) even when identifiable. We study how incorporating data on units for which only surrogate outcomes not of primary interest are observed can increase the precision of ATE estimation. We refrain from imposing stringent surrogacy conditions, which permit surrogates as perfect replacements for the target outcome. Instead, we supplement the available, albeit limited, observations of the target outcome with abundant observations of surrogate outcomes, without any assumptions beyond unconfounded treatment assignment and missingness and corresponding overlap conditions. To quantify the potential gains, we derive the difference in efficiency bounds on ATE estimation with and without surrogates, both when an overwhelming or comparable number of units have missing outcomes. We develop robust ATE estimation and inference methods that realize these efficiency gains. We empirically demonstrate the gains by studying long-term-earning effects of job training.

Nathan Kallus, Xiaojie Mao, Masatoshi Uehara

We consider estimating a low-dimensional parameter in an estimating equation involving high-dimensional nuisances that depend on the parameter. A central example is the efficient estimating equation for the (local) quantile treatment effect ((L)QTE) in causal inference, which involves as a nuisance the covariate-conditional cumulative distribution function evaluated at the quantile to be estimated. Debiased machine learning (DML) is a data-splitting approach to estimating high-dimensional nuisances using flexible machine learning methods, but applying it to problems with parameter-dependent nuisances is impractical. For (L)QTE, DML requires we learn the whole covariate-conditional cumulative distribution function. We instead propose localized debiased machine learning (LDML), which avoids this burdensome step and needs only estimate nuisances at a single initial rough guess for the parameter. For (L)QTE, LDML involves learning just two regression functions, a standard task for machine learning methods. We prove that under lax rate conditions our estimator has the same favorable asymptotic behavior as the infeasible estimator that uses the unknown true nuisances. Thus, LDML notably enables practically-feasible and theoretically-grounded efficient estimation of important quantities in causal inference such as (L)QTEs when we must control for many covariates and/or flexible relationships, as we demonstrate in empirical studies.

Yichun Hu, Nathan Kallus, Xiaojie Mao

We study a nonparametric contextual bandit problem where the expected reward functions belong to a Hölder class with smoothness parameter $β$. We show how this interpolates between two extremes that were previously studied in isolation: non-differentiable bandits ($β\leq1$), where rate-optimal regret is achieved by running separate non-contextual bandits in different context regions, and parametric-response bandits (satisfying $β=\infty$), where rate-optimal regret can be achieved with minimal or no exploration due to infinite extrapolatability. We develop a novel algorithm that carefully adjusts to all smoothness settings and we prove its regret is rate-optimal by establishing matching upper and lower bounds, recovering the existing results at the two extremes. In this sense, our work bridges the gap between the existing literature on parametric and non-differentiable contextual bandit problems and between bandit algorithms that exclusively use global or local information, shedding light on the crucial interplay of complexity and regret in contextual bandits.

Nathan Kallus, Xiaojie Mao, Angela Zhou

The increasing impact of algorithmic decisions on people's lives compels us to scrutinize their fairness and, in particular, the disparate impacts that ostensibly-color-blind algorithms can have on different groups. Examples include credit decisioning, hiring, advertising, criminal justice, personalized medicine, and targeted policymaking, where in some cases legislative or regulatory frameworks for fairness exist and define specific protected classes. In this paper we study a fundamental challenge to assessing disparate impacts in practice: protected class membership is often not observed in the data. This is particularly a problem in lending and healthcare. We consider the use of an auxiliary dataset, such as the US census, to construct models that predict the protected class from proxy variables, such as surname and geolocation. We show that even with such data, a variety of common disparity measures are generally unidentifiable, providing a new perspective on the documented biases of popular proxy-based methods. We provide exact characterizations of the tightest-possible set of all possible true disparities that are consistent with the data (and possibly any assumptions). We further provide optimization-based algorithms for computing and visualizing these sets and statistical tools to assess sampling uncertainty. Together, these enable reliable and robust assessments of disparities -- an important tool when disparity assessment can have far-reaching policy implications. We demonstrate this in two case studies with real data: mortgage lending and personalized medicine dosing.

Jiahao Chen, Nathan Kallus, Xiaojie Mao et al.

Assessing the fairness of a decision making system with respect to a protected class, such as gender or race, is challenging when class membership labels are unavailable. Probabilistic models for predicting the protected class based on observable proxies, such as surname and geolocation for race, are sometimes used to impute these missing labels for compliance assessments. Empirically, these methods are observed to exaggerate disparities, but the reason why is unknown. In this paper, we decompose the biases in estimating outcome disparity via threshold-based imputation into multiple interpretable bias sources, allowing us to explain when over- or underestimation occurs. We also propose an alternative weighted estimator that uses soft classification, and show that its bias arises simply from the conditional covariance of the outcome with the true class membership. Finally, we illustrate our results with numerical simulations and a public dataset of mortgage applications, using geolocation as a proxy for race. We confirm that the bias of threshold-based imputation is generally upward, but its magnitude varies strongly with the threshold chosen. Our new weighted estimator tends to have a negative bias that is much simpler to analyze and reason about.

Nathan Kallus, Xiaojie Mao, Angela Zhou

We study the problem of learning conditional average treatment effects (CATE) from observational data with unobserved confounders. The CATE function maps baseline covariates to individual causal effect predictions and is key for personalized assessments. Recent work has focused on how to learn CATE under unconfoundedness, i.e., when there are no unobserved confounders. Since CATE may not be identified when unconfoundedness is violated, we develop a functional interval estimator that predicts bounds on the individual causal effects under realistic violations of unconfoundedness. Our estimator takes the form of a weighted kernel estimator with weights that vary adversarially. We prove that our estimator is sharp in that it converges exactly to the tightest bounds possible on CATE when there may be unobserved confounders. Further, we study personalized decision rules derived from our estimator and prove that they achieve optimal minimax regret asymptotically. We assess our approach in a simulation study as well as demonstrate its application in the case of hormone replacement therapy by comparing conclusions from a real observational study and clinical trial.

Nathan Kallus, Xiaojie Mao, Madeleine Udell

Valid causal inference in observational studies often requires controlling for confounders. However, in practice measurements of confounders may be noisy, and can lead to biased estimates of causal effects. We show that we can reduce the bias caused by measurement noise using a large number of noisy measurements of the underlying confounders. We propose the use of matrix factorization to infer the confounders from noisy covariates, a flexible and principled framework that adapts to missing values, accommodates a wide variety of data types, and can augment many causal inference methods. We bound the error for the induced average treatment effect estimator and show it is consistent in a linear regression setting, using Exponential Family Matrix Completion preprocessing. We demonstrate the effectiveness of the proposed procedure in numerical experiments with both synthetic data and real clinical data.