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

Abstract

The goal of this note is to give a detailed proof, to the best of our understanding, of the recent presentation by Harrison and Leeman (arXiv:2609.17650v01 and arXiv:2609.17650v02) of the proof by Astra on the lower bound for differentially private continual counting. We believe a more natural and easy proof is possible and hope that this note will help in that effort. Prior to the initial preprint by Harrison and Leeman (arXiv:2609.17650v01), Bairaktari and Larsen (arXiv:2607.00876) gave an elegant proof to show a lower bound of $\Omega(\log^{3/2}(n))$ for both pure and approximate-DP continual counting, and in personal communication had informed us that they have a proof of optimal $\Omega(\log^{2}(n))$ for pure-differential private continual counting as well. They have subsequently published their $\Omega(\log^{2}(n))$ bound, which is now a joint work of Bairaktari, Dahl, and Larsen (arXiv:2607.00876v3). Their new result is an elegant extension of their technique for approximate-differential privacy. Although the two proofs are technically different, the Astra argument uses related tree geometry introduced in Bairaktari and Larsen.