Private Incremental Regression

TL;DR

This paper addresses private incremental regression in streaming data, proposing mechanisms that maintain good empirical risk minimization under differential privacy with improved bounds for certain data geometries.

Contribution

It introduces a generic method to convert private batch ERM into incremental ERM and presents two new mechanisms for private incremental regression, improving risk bounds under specific conditions.

Findings

Baseline transformation from batch to incremental ERM.

Proposed gradient-based mechanism achieves excess risk of ~√d.

Geometric properties can improve risk bounds for specific problems.

Abstract

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 $\approx\sqrt{d}$, where $d$ 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.