252 research outputs found
Differentially Private Model Selection with Penalized and Constrained Likelihood
In statistical disclosure control, the goal of data analysis is twofold: The
released information must provide accurate and useful statistics about the
underlying population of interest, while minimizing the potential for an
individual record to be identified. In recent years, the notion of differential
privacy has received much attention in theoretical computer science, machine
learning, and statistics. It provides a rigorous and strong notion of
protection for individuals' sensitive information. A fundamental question is
how to incorporate differential privacy into traditional statistical inference
procedures. In this paper we study model selection in multivariate linear
regression under the constraint of differential privacy. We show that model
selection procedures based on penalized least squares or likelihood can be made
differentially private by a combination of regularization and randomization,
and propose two algorithms to do so. We show that our private procedures are
consistent under essentially the same conditions as the corresponding
non-private procedures. We also find that under differential privacy, the
procedure becomes more sensitive to the tuning parameters. We illustrate and
evaluate our method using simulation studies and two real data examples
Tight Lower Bounds for Differentially Private Selection
A pervasive task in the differential privacy literature is to select the
items of "highest quality" out of a set of items, where the quality of each
item depends on a sensitive dataset that must be protected. Variants of this
task arise naturally in fundamental problems like feature selection and
hypothesis testing, and also as subroutines for many sophisticated
differentially private algorithms.
The standard approaches to these tasks---repeated use of the exponential
mechanism or the sparse vector technique---approximately solve this problem
given a dataset of samples. We provide a tight lower
bound for some very simple variants of the private selection problem. Our lower
bound shows that a sample of size is required
even to achieve a very minimal accuracy guarantee.
Our results are based on an extension of the fingerprinting method to sparse
selection problems. Previously, the fingerprinting method has been used to
provide tight lower bounds for answering an entire set of queries, but
often only some much smaller set of queries are relevant. Our extension
allows us to prove lower bounds that depend on both the number of relevant
queries and the total number of queries
Private Incremental Regression
Data is continuously generated by modern data sources, and a recent challenge
in machine learning has been to develop techniques that perform well in an
incremental (streaming) setting. In this paper, we investigate the problem of
private machine learning, where as common in practice, the data is not given at
once, but rather arrives incrementally over time.
We introduce the problems of private incremental ERM and private incremental
regression where the general goal is to always maintain a good empirical risk
minimizer for the history observed under differential privacy. Our first
contribution is a generic transformation of private batch ERM mechanisms into
private incremental ERM mechanisms, based on a simple idea of invoking the
private batch ERM procedure at some regular time intervals. We take this
construction as a baseline for comparison. We then provide two mechanisms for
the private incremental regression problem. Our first mechanism is based on
privately constructing a noisy incremental gradient function, which is then
used in a modified projected gradient procedure at every timestep. This
mechanism has an excess empirical risk of , where is the
dimensionality of the data. While from the results of [Bassily et al. 2014]
this bound is tight in the worst-case, we show that certain geometric
properties of the input and constraint set can be used to derive significantly
better results for certain interesting regression problems.Comment: To appear in PODS 201
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