# Lap2: Revisiting Laplace DP-SGD for High Dimensions via Majorization Theory

arXiv: https://arxiv.org/abs/2602.23516 · HTML: https://mlsanity.com/paper/2602.23516/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 41 of its 66 references that are in the map.
- First posted on arXiv 26 Feb 2026.

## Builds on (most cited first, 30 of 41)

- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [RoBERTa: A Robustly Optimized BERT Pretraining Approach](https://mlsanity.com/paper/1907.11692/) (2019)
- [GLUE: A Multi-Task Benchmark and Analysis Platform for Natural Language Understanding](https://mlsanity.com/paper/1804.07461/) (2018)
- [Deep Learning with Differential Privacy](https://mlsanity.com/paper/1607.00133/) (2016)
- [The Secret Sharer: Evaluating and Testing Unintended Memorization in Neural Networks](https://mlsanity.com/paper/1802.08232/) (2018)
- [Inverting Gradients -- How easy is it to break privacy in federated learning?](https://mlsanity.com/paper/2003.14053/) (2020)
- [Large Language Models Can Be Strong Differentially Private Learners](https://mlsanity.com/paper/2110.05679/) (2021)
- [Differentially Private Fine-tuning of Language Models](https://mlsanity.com/paper/2110.06500/) (2021)
- [Opacus: User-Friendly Differential Privacy Library in PyTorch](https://mlsanity.com/paper/2109.12298/) (2021)
- [Laplace Redux -- Effortless Bayesian Deep Learning](https://mlsanity.com/paper/2106.14806/) (2021)
- [The Composition Theorem for Differential Privacy](https://mlsanity.com/paper/1311.0776/) (2013)
- [Subsampled R'enyi Differential Privacy and Analytical Moments Accountant](https://mlsanity.com/paper/1808.00087/) (2018)
- [Unlocking High-Accuracy Differentially Private Image Classification through Scale](https://mlsanity.com/paper/2204.13650/) (2022)
- [Improving the Gaussian Mechanism for Differential Privacy: Analytical Calibration and Optimal Denoising](https://mlsanity.com/paper/1805.06530/) (2018)
- [R'enyi Differential Privacy of the Sampled Gaussian Mechanism](https://mlsanity.com/paper/1908.10530/) (2019)
- [Numerical Composition of Differential Privacy](https://mlsanity.com/paper/2106.02848/) (2021)
- [Updates-Leak: Data Set Inference and Reconstruction Attacks in Online Learning](https://mlsanity.com/paper/1904.01067/) (2019)
- [Tempered Sigmoid Activations for Deep Learning with Differential Privacy](https://mlsanity.com/paper/2007.14191/) (2020)
- [Hypothesis Testing Interpretations and Renyi Differential Privacy](https://mlsanity.com/paper/1905.09982/) (2019)
- [DP-Forward: Fine-tuning and Inference on Language Models with Differential Privacy in Forward Pass](https://mlsanity.com/paper/2309.06746/) (2023)
- [Differentially Private Optimization on Large Model at Small Cost](https://mlsanity.com/paper/2210.00038/) (2022)
- [Mixed Differential Privacy in Computer Vision](https://mlsanity.com/paper/2203.11481/) (2022)
- [Connect the Dots: Tighter Discrete Approximations of Privacy Loss Distributions](https://mlsanity.com/paper/2207.04380/) (2022)
- [TAN Without a Burn: Scaling Laws of DP-SGD](https://mlsanity.com/paper/2210.03403/) (2022)
- [Exploring the Limits of Differentially Private Deep Learning with Group-wise Clipping](https://mlsanity.com/paper/2212.01539/) (2022)
- [Scalable and Efficient Training of Large Convolutional Neural Networks with Differential Privacy](https://mlsanity.com/paper/2205.10683/) (2022)
- [Differentially Private Bias-Term Fine-tuning of Foundation Models](https://mlsanity.com/paper/2210.00036/) (2022)
- [Fast and Memory Efficient Differentially Private-SGD via JL Projections](https://mlsanity.com/paper/2102.03013/) (2021)
- [ViP: A Differentially Private Foundation Model for Computer Vision](https://mlsanity.com/paper/2306.08842/) (2023)
- [Differentially Private SGD Without Clipping Bias: An Error-Feedback Approach](https://mlsanity.com/paper/2311.14632/) (2023)

## Abstract (arXiv)

Differentially Private Stochastic Gradient Descent (DP-SGD) is a cornerstone technique for ensuring privacy in deep learning, widely used in both training from scratch and fine-tuning large-scale language models. While DP-SGD predominantly relies on the Gaussian mechanism, the Laplace mechanism remains underutilized due to its reliance on L1 norm clipping. This constraint severely limits its practicality in high-dimensional models because the L1 norm of an n-dimensional gradient can be up to sqrt(n) times larger than its L2 norm. As a result, the required noise scale grows significantly with model size, leading to poor utility or untrainable models. In this work, we introduce Lap2, a new solution that enables L2 clipping for Laplace DP-SGD while preserving strong privacy guarantees. We overcome the dimensionality-driven clipping barrier by computing coordinate-wise moment bounds and applying majorization theory to construct a tight, data-independent upper bound over the full model. By exploiting the Schur-convexity of the moment accountant function, we aggregate these bounds using a carefully designed majorization set that respects the L2 clipping constraint. This yields a multivariate privacy accountant that scales gracefully with model dimension and enables the use of thousands of moments. Empirical evaluations demonstrate that our approach significantly improves the performance of Laplace DP-SGD, achieving results comparable to or better than Gaussian DP-SGD under strong privacy constraints. For instance, fine-tuning RoBERTa-base (125M parameters) on SST-2 achieves 87.88% accuracy at epsilon=0.54, outperforming Gaussian (87.16%) and standard Laplace (48.97%) under the same budget.