Achieving local differential privacy in distributed optimization while maintaining low communication cost remains challenging. Existing vector quantization methods, such as vqSGD, use high-dimensional geometric constructions but incur unfavorable dimension-dependent variance. In this work, we propose Subsampled Stochastic TurboQuant (SSTQ), a framework that combines overcomplete equal-norm tight frames, coordinate subsampling, and privacy-aware one-dimensional quantization. SSTQ includes two variants: a Flat Randomized Response version and a Metric-Aware Laplace version, the latter being better suited to higher codebook bit-width regimes. We show that SSTQ achieves optimal mean squared error scaling while using only $\lceil \log_2 N \rceil + b$ bits per client, where $N = \Theta(d)$ is the frame size. We also derive a surrogate privacy-aware codebook objective that reduces the codebook-dependent MSE scaling from $O(4^b)$ to $O(2^b)$. Finally, we empirically evaluate SSTQ against established baselines on federated learning tasks using CIFAR-10 and Fashion-MNIST, demonstrating favorable utility and communication efficiency. Some of the analytical derivations were first obtained using a fully automated Gemini-based agentic system developed internally at Google. The authors have verified those derivations and edited them for clarity of presentation.
Adel Javanmard, David P. Woodruff, V. Mirrokni· 0 citations
For the $n\times n$ lower-triangular all-ones matrix $Q$, we prove a near-optimal lower bound \[ \gamma_{2,1}(Q) := \inf_{Q=AB} \|A\|_{2\to\infty}\|B\|_{1\to1} = \Omega\!\left( \frac{\log^{3/2}n}{(\log\log n)^{3/2}} \right), \] where the infimum ranges over real factorizations of arbitrary finite inner dimension. This cost is a central parameter in space bounds for factorization-based rank and quantile estimation in turnstile streams and in error bounds for matrix mechanisms for continual counting under pure differential privacy. The proof combines right-sided Haar projections with a scale-dependent numerical-sparsity decomposition of the rows of $B$. At each scale, a rank--Frobenius argument shows that the numerically sparse rows cannot account for all of the required Schatten $2/3$ mass, while a Haar projection estimate bounds the contribution of the remaining rows. Summing these bounds over the dyadic scales yields the result. The proof was obtained using a fully automated Gemini-based agentic system developed internally at Google. The authors verified the proof and made minor revisions.
Honghao Lin, V. Mirrokni, David P. Woodruff· 0 citations
We introduce TCS-Bench, a benchmark for evaluating Large Language Models (LLMs) on research-level Theoretical Computer Science (TCS) proof generation. TCS-Bench consists of theorem-proving tasks from papers published at top theoretical computer science venues (STOC, FOCS, and SODA). Each task provides the necessary context to derive a self-contained proof for a target result. We evaluate state-of-the-art models on this benchmark. We verify the correctness of generated proofs via a verification agent, and further benchmark the verifier against human-expert proof judgements on a set of target statements and generated proofs pairs. Our reference verifier achieves over 90% accuracy on the expert labeled set.
Vincent Cohen-Addad, Dimitris Paparas, E. V. Wijland et al.· 1 citation
This work eliminates the residual-stage $O(d)$-bit payload and reduces the leading upper-bound constant by a factor of approximately $5.93$ compared with the two-stage construction of Feng et al.
Honghao Lin, V. Mirrokni, David P. Woodruff· 0 citations
The Paper Assistant Tool is introduced, an agentic AI framework built for deep scientific review and verification and able to identify deeper issues than a single model call alone, achieving a 34% improvement over zero-shot recall on mathematical errors in the SPOT benchmark.
Rajesh Jayaram, Drew Tyler, David P. Woodruff et al.· arXiv.org· 0 citations