Author name cluster

Vikrant Singhal

Possible papers associated with this exact author name in Arrow. This page groups case-insensitive exact name matches and is not a full identity disambiguation profile.

6 papers

2 author rows

NeurIPS Conference 2023 Conference Paper

Private Distribution Learning with Public Data: The View from Sample Compression

Shai Ben-David
Alex Bie
Clément L Canonne
Gautam Kamath
Vikrant Singhal

We study the problem of private distribution learning with access to public data. In this setup, which we refer to as *public-private learning*, the learner is given public and private samples drawn from an unknown distribution $p$ belonging to a class $\mathcal Q$, with the goal of outputting an estimate of $p$ while adhering to privacy constraints (here, pure differential privacy) only with respect to the private samples. We show that the public-private learnability of a class $\mathcal Q$ is connected to the existence of a sample compression scheme for $\mathcal Q$, as well as to an intermediate notion we refer to as \emph{list learning}. Leveraging this connection: (1) approximately recovers previous results on Gaussians over $\mathbb R^d$; and (2) leads to new ones, including sample complexity upper bounds for arbitrary $k$-mixtures of Gaussians over $\mathbb R^d$, results for agnostic and distribution-shift resistant learners, as well as closure properties for public-private learnability under taking mixtures and products of distributions. Finally, via the connection to list learning, we show that for Gaussians in $\mathbb R^d$, at least $d$ public samples are necessary for private learnability, which is close to the known upper bound of $d+1$ public samples.

PDF Details

NeurIPS Conference 2022 Conference Paper

New Lower Bounds for Private Estimation and a Generalized Fingerprinting Lemma

Gautam Kamath
Argyris Mouzakis
Vikrant Singhal

We prove new lower bounds for statistical estimation tasks under the constraint of $(\varepsilon, \delta)$-differential privacy. First, we provide tight lower bounds for private covariance estimation of Gaussian distributions. We show that estimating the covariance matrix in Frobenius norm requires $\Omega(d^2)$ samples, and in spectral norm requires $\Omega(d^{3/2})$ samples, both matching upper bounds up to logarithmic factors. We prove these bounds via our main technical contribution, a broad generalization of the fingerprinting method to exponential families. Additionally, using the private Assouad method of Acharya, Sun, and Zhang, we show a tight $\Omega(d/(\alpha^2 \varepsilon))$ lower bound for estimating the mean of a distribution with bounded covariance to $\alpha$-error in $\ell_2$-distance. Prior known lower bounds for all these problems were either polynomially weaker or held under the stricter condition of $(\varepsilon, 0)$-differential privacy.

PDF Details

NeurIPS Conference 2022 Conference Paper

Private Estimation with Public Data

Alex Bie
Gautam Kamath
Vikrant Singhal

We initiate the study of differentially private (DP) estimation with access to a small amount of public data. For private estimation of $d$-dimensional Gaussians, we assume that the public data comes from a Gaussian that may have vanishing similarity in total variation distance with the underlying Gaussian of the private data. We show that under the constraints of pure or concentrated DP, $d+1$ public data samples are sufficient to remove any dependence on the range parameters of the private data distribution from the private sample complexity, which is known to be otherwise necessary without public data. For separated Gaussian mixtures, we assume that the underlying public and private distributions are the same, and we consider two settings: (1) when given a dimension-independent amount of public data, the private sample complexity can be improved polynomially in terms of the number of mixture components, and any dependence on the range parameters of the distribution can be removed in the approximate DP case; (2) when given an amount of public data linear in the dimension, the private sample complexity can be made independent of range parameters even under concentrated DP, and additional improvements can be made to the overall sample complexity.

PDF Details

NeurIPS Conference 2021 Conference Paper

Privately Learning Subspaces

Vikrant Singhal
Thomas Steinke

Private data analysis suffers a costly curse of dimensionality. However, the data often has an underlying low-dimensional structure. For example, when optimizing via gradient descent, the gradients often lie in or near a low-dimensional subspace. If that low-dimensional structure can be identified, then we can avoid paying (in terms of privacy or accuracy) for the high ambient dimension. We present differentially private algorithms that take input data sampled from a low-dimensional linear subspace (possibly with a small amount of error) and output that subspace (or an approximation to it). These algorithms can serve as a pre-processing step for other procedures.

PDF Details

NeurIPS Conference 2019 Conference Paper

Differentially Private Algorithms for Learning Mixtures of Separated Gaussians

Gautam Kamath
Or Sheffet
Vikrant Singhal
Jonathan Ullman

Learning the parameters of Gaussian mixture models is a fundamental and widely studied problem with numerous applications. In this work, we give new algorithms for learning the parameters of a high-dimensional, well separated, Gaussian mixture model subject to the strong constraint of differential privacy. In particular, we give a differentially private analogue of the algorithm of Achlioptas and McSherry. Our algorithm has two key properties not achieved by prior work: (1) The algorithm’s sample complexity matches that of the corresponding non-private algorithm up to lower order terms in a wide range of parameters. (2) The algorithm requires very weak a priori bounds on the parameters of the mixture components.

PDF Details

STOC Conference 2016 Conference Paper

Deterministic and probabilistic binary search in graphs

Ehsan Emamjomeh-Zadeh
David Kempe 0001
Vikrant Singhal

We consider the following natural generalization of Binary Search: in a given undirected, positively weighted graph, one vertex is a target . The algorithm’s task is to identify the target by adaptively querying vertices. In response to querying a node q , the algorithm learns either that q is the target, or is given an edge out of q that lies on a shortest path from q to the target. We study this problem in a general noisy model in which each query independently receives a correct answer with probability p > 1/2 (a known constant), and an (adversarial) incorrect one with probability 1 − p . Our main positive result is that when p = 1 (i.e., all answers are correct), log 2 n queries are always sufficient. For general p , we give an (almost information-theoretically optimal) algorithm that uses, in expectation, no more than (1 − δ) log n /1 − H ( p ) + o (log n ) + O (log 2 (1/δ)) queries, and identifies the target correctly with probability at leas 1 − δ. Here, H ( p ) = −( p log p + (1 − p ) log(1 − p )) denotes the entropy. The first bound is achieved by the algorithm that iteratively queries a 1-median of the nodes not ruled out yet; the second bound by careful repeated invocations of a multiplicative weights algorithm. Even for p = 1, we show several hardness results for the problem of determining whether a target can be found using K queries. Our upper bound of log 2 n implies a quasipolynomial-time algorithm for undirected connected graphs; we show that this is best-possible under the Strong Exponential Time Hypothesis (SETH). Furthermore, for directed graphs, or for undirected graphs with non-uniform node querying costs, the problem is PSPACE-complete. For a semi-adaptive version, in which one may query r nodes each in k rounds, we show membership in Σ 2 k −1 in the polynomial hierarchy, and hardness for Σ 2 k −5 .

Details