Mean Estimation with User-level Privacy under Data Heterogeneity

A key challenge in many modern data analysis tasks is that user data are heterogeneous. Different users may possess vastly different numbers of data points. More importantly, it cannot be assumed that all users sample from the same underlying distribution. This is true, for example in language data, where different speech styles result in data heterogeneity. In this work we propose a simple model of heterogeneous user data that allows user data to differ in both distribution and quantity of data, and provide a method for estimating the population-level mean while preserving user-level differential privacy. We demonstrate asymptotic optimality of our estimator and also prove general lower bounds on the error achievable in the setting we introduce.

Differentially Private Synthetic Control

Synthetic control is a causal inference tool used to estimate the treatment effects of an intervention by creating synthetic counterfactual data. This approach combines measurements from other similar observations (i.e., donor pool ) to predict a counterfactual time series of interest (i.e., target unit) by analyzing the relationship between the target and the donor pool before the intervention. As synthetic control tools are increasingly applied to sensitive or proprietary data, formal privacy protections are often required. In this work, we provide the first algorithms for differentially private synthetic control with explicit error bounds. Our approach builds upon tools from non-private synthetic control and differentially private empirical risk minimization. We provide upper and lower bounds on the sensitivity of the synthetic control query and provide explicit error bounds on the accuracy of our private synthetic control algorithms. We show that our algorithms produce accurate predictions for the target unit, and that the cost of privacy is small. Finally, we empirically evaluate the performance of our algorithm, and show favorable performance in a variety of parameter regimes, as well as providing guidance to practitioners for hyperparameter tuning.

Robust Estimation under the Wasserstein Distance

We study the problem of robust distribution estimation under the Wasserstein metric, a popular discrepancy measure between probability distributions rooted in optimal transport (OT) theory. We introduce a new outlier-robust Wasserstein distance $\mathsf{W}_p^\varepsilon$ which allows for $\varepsilon$ outlier mass to be removed from its input distributions, and show that minimum distance estimation under $\mathsf{W}_p^\varepsilon$ achieves minimax optimal robust estimation risk. Our analysis is rooted in several new results for partial OT, including an approximate triangle inequality, which may be of independent interest. To address computational tractability, we derive a dual formulation for $\mathsf{W}_p^\varepsilon$ that adds a simple penalty term to the classic Kantorovich dual objective. As such, $\mathsf{W}_p^\varepsilon$ can be implemented via an elementary modification to standard, duality-based OT solvers. Our results are extended to sliced OT, where distributions are projected onto low-dimensional subspaces, and applications to homogeneity and independence testing are explored. We illustrate the virtues of our framework via applications to generative modeling with contaminated datasets.

Private Sequential Hypothesis Testing for Statisticians: Privacy, Error Rates, and Sample Size

The sequential hypothesis testing problem is a class of statistical analyses where the sample size is not fixed in advance. Instead, the decision-process takes in new observations sequentially to make real-time decisions for testing an alternative hypothesis against a null hypothesis until some stopping criterion is satisfied. In many common applications of sequential hypothesis testing, the data can be highly sensitive and may require privacy protection; for example, sequential hypothesis testing is used in clinical trials, where doctors sequentially collect data from patients and must determine when to stop recruiting patients and whether the treatment is effective. The field of differential privacy has been developed to offer data analysis tools with strong privacy guarantees, and has been commonly applied to machine learning and statistical tasks. In this work, we study the sequential hypothesis testing problem under a slight variant of differential privacy, known as Renyi differential privacy. We present a new private algorithm based on Wald's Sequential Probability Ratio Test (SPRT) that also gives strong theoretical privacy guarantees. We provide theoretical analysis on statistical performance measured by Type I and Type II error as well as the expected sample size. We also empirically validate our theoretical results on several synthetic databases, showing that our algorithms also perform well in practice. Unlike previous work in private hypothesis testing that focused only on the classical fixed sample setting, our results in the sequential setting allow a conclusion to be reached much earlier, and thus saving the cost of collecting additional samples.

Outlier-Robust Optimal Transport: Duality, Structure, and Statistical Analysis

The Wasserstein distance, rooted in optimal transport (OT) theory, is a popular discrepancy measure between probability distributions with various applications to statistics and machine learning. Despite their rich structure and demonstrated utility, Wasserstein distances are sensitive to outliers in the considered distributions, which hinders applicability in practice. Inspired by the Huber contamination model, we propose a new outlier-robust Wasserstein distance $\mathsf{W}_p^\varepsilon$ which allows for $\varepsilon$ outlier mass to be removed from each contaminated distribution. Our formulation amounts to a highly regular optimization problem that lends itself better for analysis compared to previously considered frameworks. Leveraging this, we conduct a thorough theoretical study of $\mathsf{W}_p^\varepsilon$, encompassing characterization of optimal perturbations, regularity, duality, and statistical estimation and robustness results. In particular, by decoupling the optimization variables, we arrive at a simple dual form for $\mathsf{W}_p^\varepsilon$ that can be implemented via an elementary modification to standard, duality-based OT solvers. We illustrate the benefits of our framework via applications to generative modeling with contaminated datasets.

Outlier-Robust Optimal Transport: Duality, Structure, and Statistical Applications

The Wasserstein distance, rooted in optimal transport (OT) theory, is a popular discrepancy measure between probability distributions with various applications to statistics and machine learning. Despite their rich structure and demonstrated utility, Wasserstein distances are sensitive to outliers in the considered distributions, which hinders applicability in practice. Inspired by the Huber contamination model, we propose a new outlier-robust Wasserstein distance $\mathsf{W}_p^\varepsilon$ which allows for $\varepsilon$ outlier mass to be removed from each contaminated distribution. Our formulation amounts to a highly regular optimization problem that lends itself better for analysis compared to previously considered frameworks. Leveraging this, we conduct a thorough theoretical study of $\mathsf{W}_p^\varepsilon$, encompassing characterization of optimal perturbations, regularity, duality, and statistical estimation and robustness results. In particular, by decoupling the optimization variables, we arrive at a simple dual form for $\mathsf{W}_p^\varepsilon$ that can be implemented via an elementary modification to standard, duality-based OT solvers. We illustrate the benefits of our framework via applications to generative modeling with contaminated datasets.

Differentially Private Normalizing Flows for Privacy-Preserving Density Estimation

Normalizing flow models have risen as a popular solution to the problem of density estimation, enabling high-quality synthetic data generation as well as exact probability density evaluation. However, in contexts where individuals are directly associated with the training data, releasing such a model raises privacy concerns. In this work, we propose the use of normalizing flow models that provide explicit differential privacy guarantees as a novel approach to the problem of privacy-preserving density estimation. We evaluate the efficacy of our approach empirically using benchmark datasets, and we demonstrate that our method substantially outperforms previous state-of-the-art approaches. We additionally show how our algorithm can be applied to the task of differentially private anomaly detection.

Differentially Private Online Submodular Maximization

In this work we consider the problem of online submodular maximization under a cardinality constraint with differential privacy (DP). A stream of $T$ submodular functions over a common finite ground set $U$ arrives online, and at each time-step the decision maker must choose at most $k$ elements of $U$ before observing the function. The decision maker obtains a payoff equal to the function evaluated on the chosen set, and aims to learn a sequence of sets that achieves low expected regret. In the full-information setting, we develop an $(\varepsilon,\delta)$-DP algorithm with expected $(1-1/e)$-regret bound of $\mathcal{O}\left( \frac{k^2\log |U|\sqrt{T \log k/\delta}}{\varepsilon} \right)$. This algorithm contains $k$ ordered experts that learn the best marginal increments for each item over the whole time horizon while maintaining privacy of the functions. In the bandit setting, we provide an $(\varepsilon,\delta+ O(e^{-T^{1/3}}))$-DP algorithm with expected $(1-1/e)$-regret bound of $\mathcal{O}\left( \frac{\sqrt{\log k/\delta}}{\varepsilon} (k (|U| \log |U|)^{1/3})^2 T^{2/3} \right)$. Our algorithms contains $k$ ordered experts that learn the best marginal item to select given the items chosen her predecessors, while maintaining privacy of the functions. One challenge for privacy in this setting is that the payoff and feedback of expert $i$ depends on the actions taken by her $i-1$ predecessors. This particular type of information leakage is not covered by post-processing, and new analysis is required. Our techniques for maintaining privacy with feedforward may be of independent interest.

Attribute Privacy: Framework and Mechanisms

Ensuring the privacy of training data is a growing concern since many machine learning models are trained on confidential and potentially sensitive data. Much attention has been devoted to methods for protecting individual privacy during analyses of large datasets. However in many settings, global properties of the dataset may also be sensitive (e.g., mortality rate in a hospital rather than presence of a particular patient in the dataset). In this work, we depart from individual privacy to initiate the study of attribute privacy, where a data owner is concerned about revealing sensitive properties of a whole dataset during analysis. We propose definitions to capture \emph{attribute privacy} in two relevant cases where global attributes may need to be protected: (1) properties of a specific dataset and (2) parameters of the underlying distribution from which dataset is sampled. We also provide two efficient mechanisms and one inefficient mechanism that satisfy attribute privacy for these settings. We base our results on a novel use of the Pufferfish framework to account for correlations across attributes in the data, thus addressing "the challenging problem of developing Pufferfish instantiations and algorithms for general aggregate secrets" that was left open by \cite{kifer2014pufferfish}.

Locally Interpretable Predictions of Parkinson's Disease Progression

In precision medicine, machine learning techniques have been commonly proposed to aid physicians in early screening of chronic diseases such as Parkinson's Disease. These automated screening procedures should be interpretable by a clinician who must explain the decision-making process to patients for informed consent. However, the methods which typically achieve the highest level of accuracy given early screening data are complex black box models. In this paper, we provide a novel approach for explaining black box model predictions of Parkinson's Disease progression that can give high fidelity explanations with lower model complexity. Specifically, we use the Parkinson's Progression Marker Initiative (PPMI) data set to cluster patients based on the trajectory of their disease progression. This can be used to predict how a patient's symptoms are likely to develop based on initial screening data. We then develop a black box (random forest) model for predicting which cluster a patient belongs in, along with a method for generating local explainers for these predictions. Our local explainer methodology uses a computationally efficient information filter to include only the most relevant features. We also develop a global explainer methodology and empirically validate its performance on the PPMI data set, showing that our approach may Pareto-dominate existing techniques on the trade-off between fidelity and coverage. Such tools should prove useful for implementing medical screening tools in practice by providing explainer models with high fidelity and significantly less functional complexity.