An Exposition of GPT Astra's Proof of Lower Bound on DP Continual Counting

Private continual counting is a fundamental problem in differential privacy: given a binary stream of length $n$, where each $1$ corresponds to the contribution of one individual, the goal is to release all running counts while protecting the privacy of each individual. The standard algorithm is the binary tree mechanism, whose Gaussian-noise variant achieves expected $\ell_\infty$ error proportional to $\log^{3/2} n$ for approximate differential privacy. Whether this dependence on the stream length is necessary has remained a central open problem. In this work, we resolve the dependence on $n$ by proving that every differentially private mechanism for continual counting must incur expected $\ell_\infty$ error $Ω(\log^{3/2} n)$. This shows that the binary tree mechanism is asymptotically optimal in the approximate-DP setting. As a consequence, we also obtain a largest-possible separation between hereditary discrepancy and private $\ell_\infty$ error for linear queries, showing that the known general upper bound in terms of hereditary discrepancy has the optimal dependence on the number of queries.

Memorization in large language models is measured through a zoo of definitions whose formal relations are unknown, and differential privacy (DP) is treated as a proxy against all of them at once. We pin down the exact DP constant for the two that carry the practical weight, counterfactual memorization and adaptive extraction, and show that they do not control each other. Under $f$-DP, every adaptive extraction protocol with list budget $m$ succeeds with probability at most $1-f(κ)$ for the oblivious baseline $κ$, and the bound is tight on a dense set of baselines: DP uniformly controls extraction exactly up to a threshold in how well the secret can be guessed a priori. Min-entropy certifies that baseline distribution-free, since $H_\infty\geε\log_2 e+\log_2(m/τ)$ holds extraction below a risk level $τ\le1/2$ under pure $ε$-DP for every prior, and is exact on uniform priors. On the memorization side, $f$-DP caps the counterfactual memorization of any bounded score at an advantage functional $η(f)$, equal to $\tanh(ε/2)$ under pure DP; for $k\ge2$ duplicated copies the naive $ε\mapsto kε$ bound $\tanh(kε/2)$ is unattainable, the exact constant being a closed-form staircase attained by geometric noisy counting. That cap is attained inside the local score class used in practice, and it is there that the two measures separate: one mechanism is memorized yet unextractable, another fully extractable yet exactly invisible to every loss-based score. The two-sided blind spot this opens for loss-based auditing and unlearning verification survives on billion-parameter models: a reserved-trigger release is recovered verbatim from one prompt while the audits practitioners deploy certify it clean.

Subgraph counting is a fundamental problem in graph analysis. Motivated by practical scenarios where graph analytics are performed on subgraphs induced by selected vertices -- rather than on the entire graph -- and by growing privacy concerns, we initiate the study of differentially private range subgraph counting (DPRSC). The goal is to privately count occurrences of a fixed pattern graph within induced subgraphs defined by multi-dimensional attribute ranges. Unlike classical point counting, subgraph counting is inherently nonlinear and exhibits high sensitivity: a single edge modification can affect many subgraph occurrences. We present the first efficient algorithms for DPRSC with small additive error. Our approach introduces a subgraph projection that reduces DPRSC to weighted orthogonal range counting, enabling the use of range trees and local sensitivity estimation to achieve accurate private query answering. We complement our algorithms with matching lower bounds, obtained by reducing reconstruction attacks to DPRSC and leveraging discrepancy theory. In particular, we show that any differentially private algorithm for DPRSC must incur additive error exponential in the dimension. Empirical evaluations demonstrate that our algorithms significantly outperform baseline methods in accuracy and runtime while maintaining strong privacy guarantees.