# Differential Privacy of Gradient Descent on Perturbed Objectives

arXiv: https://arxiv.org/abs/2610.02716 · HTML: https://mlsanity.com/paper/2610.02716/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 30 references that are in the map.
- First posted on arXiv 2 Oct 2026.

## Builds on (most cited first, 13 of 13)

- [Privacy Amplification by Subsampling: Tight Analyses via Couplings and Divergences](https://mlsanity.com/paper/1807.01647/) (2018)
- [Private Stochastic Convex Optimization with Optimal Rates](https://mlsanity.com/paper/1908.09970/) (2019)
- [Private Stochastic Convex Optimization: Optimal Rates in Linear Time](https://mlsanity.com/paper/2005.04763/) (2020)
- [The Limit Points of (Optimistic) Gradient Descent in Min-Max Optimization](https://mlsanity.com/paper/1807.03907/) (2018)
- [Privacy Amplification by Iteration](https://mlsanity.com/paper/1808.06651/) (2018)
- [Differential Privacy Dynamics of Langevin Diffusion and Noisy Gradient Descent](https://mlsanity.com/paper/2102.05855/) (2021)
- [Differentially Private Learning Needs Hidden State (Or Much Faster Convergence)](https://mlsanity.com/paper/2203.05363/) (2022)
- [Shifted Interpolation for Differential Privacy](https://mlsanity.com/paper/2403.00278/) (2024)
- [Improving the Privacy and Practicality of Objective Perturbation for Differentially Private Linear Learners](https://mlsanity.com/paper/2401.00583/) (2023)
- [Private Gradient Descent for Linear Regression: Tighter Error Bounds and Instance-Specific Uncertainty Estimation](https://mlsanity.com/paper/2402.13531/) (2024)
- [Convergent Privacy Loss of Noisy-SGD without Convexity and Smoothness](https://mlsanity.com/paper/2410.01068/) (2024)
- [Bilevel gradient methods and the Morse parametric qualification condition](https://mlsanity.com/paper/2502.09074/) (2025)
- [Quadratic Objective Perturbation: Curvature-Based Differential Privacy](https://mlsanity.com/paper/2605.05905/) (2026)

## Abstract (arXiv)

Objective perturbation adds a random linear term to a regularized empirical risk and releases the exact perturbed minimizer. We study the finite computation obtained by releasing the N-th iterate of deterministic gradient descent on wmapsto F(w;S)+langle z,wrangle, where z~mathcal N(0,σ2Id) is drawn once before optimization. For strongly convex and smooth objectives with Lipschitz Hessian, we prove an explicit condition under which the map zmapsto wN is a C1-diffeomorphism on the bounded domains used in the privacy argument, with a quantitative lower bound on the smallest singular value of its Jacobian. This permits a direct change-of-variables analysis of the finite iterate. For generalized linear models, the resulting privacy-profile bound has no explicit ambient-dimension factor once the iteration condition holds, and its finite-iteration correction decreases geometrically. By letting the free truncation parameter grow slowly with N, we recover the corresponding exact-minimizer certificate in the limit. We also bound the expected excess empirical risk by dσ2/(2μ) plus a geometrically decreasing optimization term, and transfer the result to population risk without an additional multiplicative condition-number factor in the leading statistical terms.