Learning with Locally Private Examples by Inverse Weierstrass Private Stochastic Gradient Descent

TL;DR

This paper introduces IWP-SGD, a bias-corrected stochastic gradient descent method for binary classification under local differential privacy, achieving unbiased estimates and convergence guarantees.

Contribution

It develops an inverse Weierstrass transform approach for bias correction and proposes a novel private SGD algorithm with proven convergence.

Findings

IWP-SGD converges at a rate of O(1/n).

The method effectively reduces bias in private data analysis.

Empirical results validate the approach on synthetic and real datasets.

Abstract

Releasing data once and for all under noninteractive Local Differential Privacy (LDP) enables complete data reusability, but the resulting noise may create bias in subsequent analyses. In this work, we leverage the Weierstrass transform to characterize this bias in binary classification. We prove that inverting this transform leads to a bias-correction method to compute unbiased estimates of nonlinear functions on examples released under LDP. We then build a novel stochastic gradient descent algorithm called Inverse Weierstrass Private SGD (IWP-SGD). It converges to the true population risk minimizer at a rate of $\mathcal{O}(1/n)$, with $n$ the number of examples. We empirically validate IWP-SGD on binary classification tasks using synthetic and real-world datasets.