Efficient Agnostic Learning of Univariate Densities Under Local Differential Privacy

We show that every class of distributions that is learnable up to arbitrarily small target error in total variation distance with finitely many samples is also learnable under ($\varepsilon, δ$)-differential privacy up to arbitrarily small target error with finitely many samples. This proves a conjecture of Ashtiani. Previous work by Bun, Kamath, Mouzakis, and Singhal [arXiv:2402.00267] gave a weak refutation of this conjecture, showing that this implication is not true when learnability is defined with respect to a fixed target error. In contrast, we show that, under the standard definition of learnability, the conjecture is true.

Differential privacy (DP) in machine learning is typically achieved by adding noise to model parameters (private learning) or to model outputs (private prediction). Recent work uses formal methods, namely abstract interpretation, to provide tighter privacy guarantees, but only for private prediction in classification settings. In this work, we investigate the use of formal methods as a general tool for tighter privacy analysis. First, we generalize the abstract gradient training (AGT) framework to private prediction in continuous, unbounded regression. Second, by reducing learning in parameterized models to a regression problem over the parameter space, we introduce Abstract Gradient Sampling (AGS), an algorithm that enables reachability-based analysis to provide guarantees for private learning. In both private prediction and private learning, we provide tightened privacy accounting for the AGT framework and a theoretical analysis demonstrating when our smooth sensitivity upper-bounds yield favourable privacy-utility trade-off. In practice, we validate that our regression bounds are tighter than global-sensitivity baselines on regression benchmarks, and, notably, yield the first finite privacy guarantees in settings where global prediction sensitivity is a priori unbounded. We also find that under matched conditions, our private learning algorithm can outperform standard private learners.

We study learning a discrete distribution under person-level differential privacy in the batched setting. Given $n$ users, each contributing $m$ i.i.d. samples from an unknown distribution $μ$ over $[k]$, we seek an estimate of $μ$ with total-variation error $α$. We give polynomial-time algorithms under both pure and approximate differential privacy. Under pure $(\varepsilon,0)$-DP, we achieve $n = \widetilde O(k/(α^2 m)+k/(α\sqrt{m}\varepsilon)+k/\varepsilon)$, the first polynomial-time pure-DP guarantee matching the information-theoretic optimum up to logarithmic factors. Under approximate $(\varepsilon,δ)$-DP, we achieve $n = \widetilde O(k/(α^2 m)+k/(α\sqrt{m}\varepsilon)+k+\log(1/δ)/\varepsilon)$. At $δ=2^{-Θ(k)}$, this is $\widetilde O(k)$ for fixed $α,\varepsilon,m$ and failure probability. Our algorithms use a new degree-6 sum-of-squares estimator for learning from untrusted batches. Its key ingredient is a trace-bounded diagonal majorizer for the difference between the empirical covariance of the repaired batch averages and the multinomial covariance predicted by their mean. We prove that clean data admit this certificate through semidefinite-programming duality and a new application of the stronger symmetric form of Grothendieck's inequality due to Friedland and Lim. The certificate preserves the mean-dependent variance of multinomial batch averages, yielding the near-optimal robust rate $O(η\sqrt{\log(1/η)/m})$ and providing the score required by the privacy reduction. We also prove a statistical query lower bound ruling out a uniformly smaller order of error under fixed polynomial SQ resource bounds.