Lower Bounds for Differential Privacy from Gaussian Width
Bibliographic record
Abstract
We study the optimal sample complexity of a given workload of linear queries under the constraints of differential privacy. The sample complexity of a query answering mechanism under error parameter alpha is the smallest n such that the mechanism answers the workload with error at most alpha on any database of size n. Following a line of research started by Hardt and Talwar [STOC 2010], we analyze sample complexity using the tools of asymptotic convex geometry. We study the sensitivity polytope, a natural convex body associated with a query workload that quantifies how query answers can change between neighboring databases. This is the information that, roughly speaking, is protected by a differentially private algorithm, and, for this reason, we expect that a "bigger" sensitivity polytope implies larger sample complexity. Our results identify the mean Gaussian width as an appropriate measure of the size of the polytope, and show sample complexity lower bounds in terms of this quantity. Our lower bounds completely characterize the workloads for which the Gaussian noise mechanism is optimal up to constants as those having asymptotically maximal Gaussian width. Our techniques also yield an alternative proof of Pisier's Volume Number Theorem which also suggests an approach to improving the parameters of the theorem.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.010 | 0.088 |
| Meta-epidemiology (narrow) | 0.003 | 0.001 |
| Meta-epidemiology (broad) | 0.004 | 0.003 |
| Bibliometrics | 0.003 | 0.003 |
| Science and technology studies | 0.002 | 0.006 |
| Scholarly communication | 0.007 | 0.019 |
| Open science | 0.005 | 0.007 |
| Research integrity | 0.003 | 0.009 |
| Insufficient payload (model declined to judge) | 0.007 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".