Mean Estimation Under Heterogeneous Privacy Demands

Differential Privacy (DP) is a well-established framework to quantify privacy loss incurred by any algorithm. Traditional formulations impose a uniform privacy requirement for all users, which is often inconsistent with real-world scenarios in which users dictate their privacy preferences individually. This work considers the problem of mean estimation, where each user can impose their own distinct privacy level. The algorithm we propose is shown to be minimax optimal and has a near-linear run-time. Our results elicit an interesting saturation phenomenon that occurs. Namely, the privacy requirements of the most stringent users dictate the overall error rates. As a consequence, users with less but differing privacy requirements are all given more privacy than they require, in equal amounts. In other words, these privacy-indifferent users are given a nontrivial degree of privacy for free, without any sacrifice in the performance of the estimator.

Worst-case vs Average-case Design for Estimation from Fixed Pairwise Comparisons

Pairwise comparison data arises in many domains, including tournament rankings, web search, and preference elicitation. Given noisy comparisons of a fixed subset of pairs of items, we study the problem of estimating the underlying comparison probabilities under the assumption of strong stochastic transitivity (SST). We also consider the noisy sorting subclass of the SST model. We show that when the assignment of items to the topology is arbitrary, these permutation-based models, unlike their parametric counterparts, do not admit consistent estimation for most comparison topologies used in practice. We then demonstrate that consistent estimation is possible when the assignment of items to the topology is randomized, thus establishing a dichotomy between worst-case and average-case designs. We propose two estimators in the average-case setting and analyze their risk, showing that it depends on the comparison topology only through the degree sequence of the topology. The rates achieved by these estimators are shown to be optimal for a large class of graphs. Our results are corroborated by simulations on multiple comparison topologies.

Denoising Linear Models with Permuted Data

The multivariate linear regression model with shuffled data and additive Gaussian noise arises in various correspondence estimation and matching problems. Focusing on the denoising aspect of this problem, we provide a characterization the minimax error rate that is sharp up to logarithmic factors. We also analyze the performance of two versions of a computationally efficient estimator, and establish their consistency for a large range of input parameters. Finally, we provide an exact algorithm for the noiseless problem and demonstrate its performance on an image point-cloud matching task. Our analysis also extends to datasets with outliers.

Linear Regression with an Unknown Permutation: Statistical and Computational Limits

Consider a noisy linear observation model with an unknown permutation, based on observing $y = \Pi^* A x^* + w$, where $x^* \in \mathbb{R}^d$ is an unknown vector, $\Pi^*$ is an unknown $n \times n$ permutation matrix, and $w \in \mathbb{R}^n$ is additive Gaussian noise. We analyze the problem of permutation recovery in a random design setting in which the entries of the matrix $A$ are drawn i.i.d. from a standard Gaussian distribution, and establish sharp conditions on the SNR, sample size $n$, and dimension $d$ under which $\Pi^*$ is exactly and approximately recoverable. On the computational front, we show that the maximum likelihood estimate of $\Pi^*$ is NP-hard to compute, while also providing a polynomial time algorithm when $d =1$.