# Quadratic Objective Perturbation: Curvature-Based Differential Privacy

arXiv: https://arxiv.org/abs/2605.05905 · HTML: https://mlsanity.com/paper/2605.05905/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 35 references that are in the map.
- First posted on arXiv 7 May 2026.

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

- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [LoRA: Low-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2106.09685/) (2021)
- [Deep Learning with Differential Privacy](https://mlsanity.com/paper/1607.00133/) (2016)
- [Benign Overfitting in Linear Regression](https://mlsanity.com/paper/1906.11300/) (2019)
- [Private Stochastic Convex Optimization with Optimal Rates](https://mlsanity.com/paper/1908.09970/) (2019)
- [Efficient Private ERM for Smooth Objectives](https://mlsanity.com/paper/1703.09947/) (2017)
- [A Three-Operator Splitting Scheme and its Optimization Applications](https://mlsanity.com/paper/1504.01032/) (2015)
- [Differentially Private Empirical Risk Minimization with Input Perturbation](https://mlsanity.com/paper/1710.07425/) (2017)
- [Privately Publishable Per-instance Privacy](https://mlsanity.com/paper/2111.02281/) (2021)
- [Improving the Privacy and Practicality of Objective Perturbation for Differentially Private Linear Learners](https://mlsanity.com/paper/2401.00583/) (2023)
- [Three Operator Splitting with Subgradients, Stochastic Gradients, and Adaptive Learning Rates](https://mlsanity.com/paper/2110.03274/) (2021)
- [Stochastic Krasnoselskii-Mann Iterations: Convergence without Uniformly Bounded Variance](https://mlsanity.com/paper/2604.22581/) (2026)
- [Stochastic Three-Composite Convex Minimization](https://mlsanity.com/paper/1701.09033/) (2017)

## Built on by (most cited first, 1 of 1)

- [Differential Privacy of Gradient Descent on Perturbed Objectives](https://mlsanity.com/paper/2610.02716/) (2026)

## Abstract (arXiv)

Objective perturbation is a standard mechanism in differentially private empirical risk minimization. In particular, Linear Objective Perturbation (LOP) enforces privacy by adding a random linear term, while strong convexity and stability are ensured by an additional deterministic quadratic term. However, this approach requires the strong assumption of bounded gradients of the loss function, which excludes many modern machine learning models. In this work, we introduce Quadratic Objective Perturbation (QOP), which perturbs the objective with a random quadratic form. This perturbation induces strong convexity and enforces stability of the problem through curvature, thereby enabling privacy and allowing sensitivity to be controlled through spectral properties of the perturbation rather than assumptions on the gradients. As a result, we obtain (varepsilon, δ)-differential privacy under weaker redgradient assumptions. Furthermore, we extend the analysis to account for approximate solutions, showing that privacy guarantees are preserved under inexact solves. Additionally, we derive utility guarantees in terms of empirical excess risk, and provide a theoretical and numerical comparison to LOP, highlighting the advantages of curvature-based perturbations. Finally, we discuss algorithmic aspects and show that the resulting problems can be solved efficiently using modern splitting schemes.