Differentially Private Ordinary Least Squares
Abstract
Linear regression is one of the most prevalent techniques in machine learning; however, it is also common to use linear regression for its explanatory capabilities rather than label prediction. Ordinary Least Squares (OLS) is often used in statistics to establish a correlation between an attribute (e.g. gender) and a label (e.g. income) in the presence of other (potentially correlated) features. OLS assumes a particular model that randomly generates the data, and derives t-values — representing the likelihood of each real value to be the true correlation. Using t-values, OLS can release a confidence interval, which is an interval on the reals that is likely to contain the true correlation; and when this interval does not intersect the origin, we can reject the null hypothesis as it is likely that the true correlation is non-zero. Our work aims at achieving similar guarantees on data under differentially private estimators. First, we show that for well-spread data, the Gaussian Johnson-Lindenstrauss Transform (JLT) gives a very good approximation of t-values; secondly, when JLT approximates Ridge regression (linear regression with $l_2$-regularization) we derive, under certain conditions, confidence intervals using the projected data; lastly, we derive, under different conditions, confidence intervals for the “Analyze Gauss” algorithm (Dwork et al 2014).
BibTeX
@InProceedings{pmlr-v70-sheffet17a,
title = {Differentially Private Ordinary Least Squares},
author = {Or Sheffet},
booktitle = {Proceedings of the 34th International Conference on Machine Learning},
pages = {3105--3114},
year = {2017},
editor = {Precup, Doina and Teh, Yee Whye},
volume = {70},
series = {Proceedings of Machine Learning Research},
month = {06--11 Aug},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v70/sheffet17a/sheffet17a.pdf},
url = {https://proceedings.mlr.press/v70/sheffet17a.html},
abstract = {Linear regression is one of the most prevalent techniques in machine learning; however, it is also common to use linear regression for its explanatory capabilities rather than label prediction. Ordinary Least Squares (OLS) is often used in statistics to establish a correlation between an attribute (e.g. gender) and a label (e.g. income) in the presence of other (potentially correlated) features. OLS assumes a particular model that randomly generates the data, and derives t-values — representing the likelihood of each real value to be the true correlation. Using t-values, OLS can release a confidence interval, which is an interval on the reals that is likely to contain the true correlation; and when this interval does not intersect the origin, we can reject the null hypothesis as it is likely that the true correlation is non-zero. Our work aims at achieving similar guarantees on data under differentially private estimators. First, we show that for well-spread data, the Gaussian Johnson-Lindenstrauss Transform (JLT) gives a very good approximation of t-values; secondly, when JLT approximates Ridge regression (linear regression with $l_2$-regularization) we derive, under certain conditions, confidence intervals using the projected data; lastly, we derive, under different conditions, confidence intervals for the “Analyze Gauss” algorithm (Dwork et al 2014).}
}