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

Applying differential privacy (DP) via DP-SGD to Low-Rank Adaptation (LoRA) is a natural approach for privacy-preserving fine-tuning. However, LoRA's low-rank parameterization poses a fundamental challenge. In LoRA, each trainable update is represented as a low-rank matrix $Z = AB^\top$, but this factorization is inherently non-identifiable: many factor pairs $(A,B)$ represent the same update $Z$. As a result, applying DP-SGD directly to the factors induces gauge-dependent perturbations on $Z$, and we show that this naive DP-LoRA can lead to unbounded noise amplification. We propose PRISM, an intrinsic DP mechanism for LoRA that is gauge invariant by construction, avoids bilinear noise amplification, and admits an efficient low-dimensional noise sampler. Moreover, PRISM yields a closed-form characterization of the effective intrinsic noise induced on $Z$, enabling stable privacy-utility trade-offs through bounded, gauge-invariant perturbations. We establish standard $(ε,δ)$-DP guarantees for PRISM and introduce a DP-aware, gauge-invariant adaptive update rule that prevents adaptive optimization from amplifying injected privacy noise, improving numerical stability in practice.

We propose a general framework for non-interactive locally differentially private (LDP) learning of structured univariate distributions that can be approximated by piece-wise polynomials. This gives the first sample-efficient LDP learners for several distribution classes, including mixtures of log-concave distributions, such as Gaussian mixture models (GMMs). These learners also provide agnostic guarantees and are computationally efficient. We also prove a lower bound for learning GMMs under LDP, improving on the standard pure-DP packing lower bounds of Bun et al. [BKSW19]. To obtain our general learning framework, we develop an efficient non-interactive LDP algorithm for synthetic data generation over large domains, which may be of independent interest. We then use it together with a deterministic adaptation of the non-private piecewise polynomial learner of Acharya et al. [ADLS17].

While Local Differential Privacy (LDP) serves as a foundational primitive for distributed data collection, its stringent randomization requirements often lead to severe degradation in data representation utility. This degradation stems from the task-agnostic nature of conventional LDP mechanisms, which perturb all dimensions without accounting for their relative importance to the downstream objective. To address this issue, we propose a novel approach that mitigates noise in task-relevant subspaces of the data representation. Our method identifies task-critical subspaces via the Jacobian of a public downstream model, selectively attenuates noise along these directions, and reshapes the isotropic noise of standard LDP mechanisms into an anisotropic distribution. The resulting mechanism preserves the privacy guarantee of the underlying LDP randomizer while heterogeneously modulating the impact of noise across task directions, thereby substantially enhancing data utility. The approach is applicable to both linear and nonlinear models and can be seamlessly integrated with existing LDP mechanisms. Extensive experiments on CIFAR-10-C under brightness corruption at the highest severity level demonstrate that integrating our approach improves classification accuracy by approximately 8 percentage points for Laplace and 20 percentage points for PrivUnit variants at $ε=7.5$. The source code is available at https://github.com/ymha/jacobian-anr-ldp.