This paper resolves two major open problems: adaptive data analysis (ADA) and adversarially robust turnstile streaming. We introduce a unified hierarchical estimator‑corrector‑learner framework. For unrestricted domains, a randomized mechanism answers $k$ adaptively chosen statistical queries with additive error $\alpha$ using only $\tilde{O}(\frac{\sqrt{k}}{\alpha} + \frac{1}{\alpha^2})$ samples, essentially matching the lower bounds of Hardt and Ullman and improving over the classic $\tilde{O}(\frac{\sqrt{k}}{\alpha^2})$ results of Dwork et al. When the domain $U$ is finite, combining the mechanism with private multiplicative weights reduces the sample complexity to $\tilde{O}\!\left(\frac{1}{\alpha^2}+ \min\!\left(\frac{\sqrt{k}}{\alpha}, \frac{\sqrt{\log |U|}}{\alpha^2}\right)\right)$.
Building on this ADA mechanism, we design a fixed linear sketch with a stateful decoder that answers about $\tilde{O}(\frac{1}{\epsilon^2})$ adaptive $F_2$ moment estimation queries within a $(1\pm\epsilon)$ multiplicative factor using $\tilde{O}(\frac{1}{\epsilon^2})$ bits of space. By embedding the sketch into a binary‑tree recursion we obtain the first nearly space‑optimal adversarially robust algorithm for $F_2$ estimation on insertion‑deletion streams. The algorithm uses $\tilde O(\frac{1}{\epsilon^2})$ bits, improving the previous $\tilde{O}(\frac{1}{\epsilon^4})$ bound of Gribelyuk et al. and matching the insertion‑only bound of Woodruff and Zhou. This shows that no extra $\epsilon$‑dependent space overhead is needed for adversarial robustness in the turnstile model.
Review: By unifying ADA and robust streaming under a single framework, the work achieves near‑optimal sample and space bounds, setting a new theoretical benchmark and offering a clear pathway for future privacy‑robust stream processing research.