Understanding Private Evolution as Learning-Augmented Clustering

We study the problem of differentially private (DP) $k$-means clustering in Euclidean space. Previous solutions rely on summing the private data directly, which induces a sensitivity proportional to the domain. We introduce PE-means, an extension of the private evolution (PE) algorithm (an increasingly popular method for synthetic data generation), to the problem of $k$-means clustering. The key advantage of PE is that it only computes a private histogram with constant sensitivity to guide the evolution. Our adaptation of PE includes new evolutionary operators for clustering, as well as other algorithmic improvements of independent interest. Overall, PE-means achieves an average improvement of 26% in clustering loss over state-of-the-art baselines such as Google's LSH-based algorithm and DP-Lloyd variants.

Generating differentially private synthetic data with meaningful Wasserstein utility guarantees is challenging in high dimensions. For datasets of size $n$ on $[0,1]^d$ with $d\ge2$, existing pure $\varepsilon$-differentially private mechanisms achieve expected $1$-Wasserstein error of order $(\varepsilon n)^{-1/d}$, reflecting the curse of dimensionality. While this rate is optimal in the worst case, it can be overly pessimistic when the data are supported on a lower-dimensional set. We formalize this through a multiscale packing-growth dimension $k$, which captures the geometric complexity of the support via the growth of packing numbers across scales. We propose \emph{Adaptive Pruned-PMM}, a pure $\varepsilon$-differentially private mechanism that combines private depth selection with our pruned variant of the Private Measure Mechanism (PMM) of He et al.\ (2023). The mechanism supports deeper, geometry-adapted hierarchies with expected running time $O\!\left(d(n+d)\log(\varepsilon n)\right)$, which is near-linear in $n$ for fixed dimension and privacy budget. Under an external multiscale packing-growth condition with dimension $k$, we show that, for fixed positive privacy budgets and fixed geometry, the expected $1$-Wasserstein error is of order $(\varepsilon n)^{-1/k}$ for $k>1$ as $n$ grows. We also prove a lower bound under a corresponding internal packing-growth condition, showing that the exponent $1/k$ is sharp within this framework.

Differentially private (DP) fine-tuning of large language models (LLMs) requires massive compute and full model access, which rules out state-of-the-art proprietary APIs for general users. Generating DP synthetic data offers a practical workaround. This approach also allows for transparent exploratory data analysis and arbitrary reuse across downstream tasks, sidestepping the rigid constraints of a model's parameter space. Private Evolution (PE) provides a promising API-based framework for generating this data, but its success relies heavily on initialization. If the private data distribution falls too far outside the foundation model's pre-training priors -- a common issue in highly specialized domain -- PE struggles to align with the target data. This misalignment causes poor convergence, degraded utility, and wasted API calls. To solve this initialization bottleneck, we introduce Metadata Augmented Private Language Evolution (MAPLE). MAPLE extracts DP tabular metadata and uses in-context learning to firmly ground the initial synthetic distribution in the target domain. Our evaluations on domain-specific text generation tasks show that MAPLE yields a strictly better privacy-utility trade-off, converges significantly faster, and sharply reduces API costs compared to baseline PE methods.