Perfect $L_2$ sampling is a powerful primitive for turnstile data streams, allowing algorithms to extract an index with probability proportional to its squared frequency, with applications to dynamic join size estimation, query optimization, heavy-hitters, sampling from self-joins, and real-time monitoring of evolving datasets. The setting $p=2$ is also the largest $p$ for which $L_p$ sampling can be achieved in polylogarithmic space. However, existing space-optimal perfect samplers rely on explicit polynomial duplication, requiring inefficient polynomial update times per stream arrival. Resolving a open question left by Swartworth, Woodruff, and Zhou [FOCS 2025], we present a perfect $L_2$ sampler that achieves an expected $\text{polylog}(n)$ update time while maintaining an optimal $\tilde{\mathcal{O}}(\log^3 n)$ bits of space.
Differential privacy (DP) has become the standard for privacy-preserving data analysis, yet its practical implementation in federated or distributed settings often faces a significant utility gap compared to the central model. While existing frameworks, including local, shuffle, and secure aggregation models, offer vetted primitives, they can incur high error for important tasks such as selection. Our first contribution is to show that this is inherent in the aggregation model: any private selection algorithm relying on the standard noisy aggregation primitive requires $Ω(\sqrt{d}/\varepsilon)$ samples. To address this limitation, we propose adding a new primitive that implements a version of the Sparse Vector Technique (SVT) and argue that it has simple implementations under various trust models. We demonstrate its power by developing an $\varepsilon$-differentially private algorithm for the selection problem. For selection in dimension $d$, our algorithm achieves an additive error of $O\left(\frac{\log(dn)}{\varepsilon}\right)$, matching that of the central model when $n$ is polynomially bounded in $d$. Furthermore, we show that this can be done using a near-optimal expected number $O(d\log^*d)$ of SVT queries. Together with existing lower bounds for the shuffle model, this result establishes an exponential separation between the SVT model and shuffle or aggregation-based models. We demonstrate the model's broad applicability to diverse tasks such as $F_p$ loss minimization, heavy-hitter identification, and sparse mean estimation.
David P. Woodruff, Vincent Cohen-Addad, Lalit Jain, Jieming Mao, Song Zuo, MohammadHossein Bateni, Simina Branzei, Michael P. Brenner 等 36 位作者
Recent advances in large language models (LLMs) have opened new avenues for accelerating scientific research. While models are increasingly capable of assisting with routine tasks, their ability to contribute to novel, expert-level mathematical discovery is less understood. We present a collection of case studies demonstrating how researchers have successfully collaborated with advanced AI models, specifically Google's Gemini-based models (in particular Gemini Deep Think and its advanced variants), to solve open problems, refute conjectures, and generate new proofs across diverse areas in theoretical computer science, as well as other areas such as economics, optimization, and physics. Based on these experiences, we extract common techniques for effective human-AI collaboration in theoretical research, such as iterative refinement, problem decomposition, and cross-disciplinary knowledge transfer. While the majority of our results stem from this interactive, conversational methodology, we also highlight specific instances that push beyond standard chat interfaces. These include deploying the model as a rigorous adversarial reviewer to detect subtle flaws in existing proofs, and embedding it within a "neuro-symbolic" loop that autonomously writes and executes code to verify complex derivations. Together, these examples highlight the potential of AI not just as a tool for automation, but as a versatile, genuine partner in the creative process of scientific discovery.