This paper studies a family of estimators based on noise-contrastive estimation (NCE) for learning unnormalized distributions. The main contribution of this work is to provide a unified perspective on various methods for learning unnormalized distributions, which have been independently proposed and studied in separate research communities, through the lens of NCE. This unified view offers new insights into existing estimators. Specifically, for exponential families, we establish the finite-sample convergence rates of the proposed estimators under a set of regularity assumptions, most of which are new.

Learning a fair predictive model is crucial to mitigate biased decisions against minority groups in high-stakes applications. A common approach to learn such a model involves solving an optimization problem that maximizes the predictive power of the model under an appropriate group fairness constraint. However, in practice, sensitive attributes are often missing or noisy resulting in uncertainty, and solely enforcing fairness constraints on uncertain sensitive attributes can fall significantly short of achieving the level of fairness without uncertainty. To understand this phenomenon, we consider the problem of fair learning for Gaussian data and reduce it to a quadratically constrained quadratic problem (QCQP). To ensure a strict fairness guarantee given uncertain sensitive attributes, we propose a robust QCQP, and characterize its solution with an intuitive geometric understanding. When uncertainty arises due to limited labeled sensitive attributes, our analysis identifies non-trivial regimes where uncertainty incurs no performance loss while continuing to guarantee strict fairness. As an illustrative example of our analysis, we propose a bootstrap-based algorithm that applies beyond the Gaussian case. We demonstrate the value of our analysis and algorithm on synthetic as well as real-world data.

Causal effect estimation from data typically requires assumptions about the cause-effect relations either explicitly in the form of a causal graph structure within the Pearlian framework, or implicitly in terms of (conditional) independence statements between counterfactual variables within the potential outcomes framework. When the treatment variable and the outcome variable are confounded, front-door adjustment is an important special case where, given the graph, causal effect of the treatment on the target can be estimated using post-treatment variables. However, the exact formula for front-door adjustment depends on the structure of the graph, which is difficult to learn in practice. In this work, we provide testable conditional independence statements to compute the causal effect using front-door-like adjustment without knowing the graph under limited structural side information. We show that our method is applicable in scenarios where knowing the Markov equivalence class is not sufficient for causal effect estimation. We demonstrate the effectiveness of our method on a class of random graphs as well as real causal fairness benchmarks.

Compressing the output of $\epsilon$-locally differentially private (LDP) randomizers naively leads to suboptimal utility. In this work, we demonstrate the benefits of using schemes that jointly compress and privatize the data using shared randomness. In particular, we investigate a family of schemes based on Minimal Random Coding (Havasi et al., 2019) and prove that they offer optimal privacy-accuracy-communication tradeoffs. Our theoretical and empirical findings show that our approach can compress PrivUnit (Bhowmick et al., 2018) and Subset Selection (Ye et al., 2018), the best known LDP algorithms for mean and frequency estimation, to the order of $\epsilon$ bits of communication while preserving their privacy and accuracy guarantees.

Treatment effect estimation from observational data is a fundamental problem in causal inference. There are two very different schools of thought that have tackled this problem. On the one hand, the Pearlian framework commonly assumes structural knowledge (provided by an expert) in the form of directed acyclic graphs and provides graphical criteria such as the back-door criterion to identify the valid adjustment sets. On the other hand, the potential outcomes (PO) framework commonly assumes that all the observed features satisfy ignorability (i.e., no hidden confounding), which in general is untestable. In prior works that attempted to bridge these frameworks, there is an observational criteria to identify an anchor variable and if a subset of covariates (not involving the anchor variable) passes a suitable conditional independence criteria, then that subset is a valid back-door. Our main result strengthens these prior results by showing that under a different expert-driven structural knowledge — that one variable is a direct causal parent of the treatment variable — remarkably, testing for subsets (not involving the known parent variable) that are valid back-doors is equivalent to an invariance test. Importantly, we also cover the non-trivial case where the entire set of observed features is not ignorable (generalizing the PO framework) without requiring the knowledge of all the parents of the treatment variable. Our key technical idea involves generation of a synthetic sub-sampling (or environment) variable that is a function of the known parent variable. In addition to designing an invariance test, this sub-sampling variable allows us to leverage Invariant Risk Minimization, and thus, connects finding valid adjustments (in non-ignorable observational settings) to representation learning. We demonstrate the effectiveness and tradeoffs of these approaches on a variety of synthetic datasets as well as real causal effect estimation benchmarks.

Selective regression allows abstention from prediction if the confidence to make an accurate prediction is not sufficient. In general, by allowing a reject option, one expects the performance of a regression model to increase at the cost of reducing coverage (i.e., by predicting on fewer samples). However, as we show, in some cases, the performance of a minority subgroup can decrease while we reduce the coverage, and thus selective regression can magnify disparities between different sensitive subgroups. Motivated by these disparities, we propose new fairness criteria for selective regression requiring the performance of every subgroup to improve with a decrease in coverage. We prove that if a feature representation satisfies the sufficiency criterion or is calibrated for mean and variance, then the proposed fairness criteria is met. Further, we introduce two approaches to mitigate the performance disparity across subgroups: (a) by regularizing an upper bound of conditional mutual information under a Gaussian assumption and (b) by regularizing a contrastive loss for conditional mean and conditional variance prediction. The effectiveness of these approaches is demonstrated on synthetic and real-world datasets.

We consider learning a sparse pairwise Markov Random Field (MRF) with continuous-valued variables from i.i.d samples. We adapt the algorithm of Vuffray et al. (2019) to this setting and provide finite-sample analysis revealing sample complexity scaling logarithmically with the number of variables, as in the discrete and Gaussian settings. Our approach is applicable to a large class of pairwise MRFs with continuous variables and also has desirable asymptotic properties, including consistency and normality under mild conditions. Further, we establish that the population version of the optimization criterion employed in Vuffray et al. (2019) can be interpreted as local maximum likelihood estimation (MLE). As part of our analysis, we introduce a robust variation of sparse linear regression a‘ la Lasso, which may be of interest in its own right.

We consider the question of learning the natural parameters of a $k$ parameter \textit{minimal} exponential family from i.i.d. samples in a computationally and statistically efficient manner. We focus on the setting where the support as well as the natural parameters are appropriately bounded. While the traditional maximum likelihood estimator for this class of exponential family is consistent, asymptotically normal, and asymptotically efficient, evaluating it is computationally hard. In this work, we propose a computationally efficient estimator that is consistent as well as asymptotically normal under mild conditions. We provide finite sample guarantees to achieve an ($\ell_2$) error of $\alpha$ in the parameter estimation with sample complexity $O(\mathrm{poly}(k/\alpha))$ and computational complexity ${O}(\mathrm{poly}(k/\alpha))$. To establish these results, we show that, at the population level, our method can be viewed as the maximum likelihood estimation of a re-parameterized distribution belonging to the same class of exponential family.