Efficient and Differentially Private Statistical Estimation via a Sum-of-Squares Exponential Mechanism
Bibliographic record
Abstract
As machine learning is applied to more privacy-sensitive data, it is becoming increasingly crucial to develop algorithms that maintain privacy. However, even the most basic high-dimensional statistical estimation tasks were not fully understood under differential privacy, specifically, there were no known efficient algorithms for mean estimation using the optimal number of samples under pure differential privacy. \n \nWe propose a new method for designing efficient and information-theoretically optimal algorithms for statistical estimation tasks that preserve privacy, using a combination of the Sum-of-Squares hierarchy and the exponential mechanism. The Sum-of-Squares hierarchy, a convex programming method, has been used to design efficient algorithms in robust statistics. The exponential mechanism, which has been widely used in differential privacy, is often used to design information-theoretically optimal algorithms, but can be inefficient. By combining these two approaches, we are able to create efficient algorithms that are also information-theoretically optimal. We apply this approach to mean estimation for heavy-tailed distributions and learning Gaussian distributions and achieve optimal results. We also show that this approach can be applied to other problems captured by the Sum-of-Squares hierarchy through a meta-theorem. Additionally, our algorithms highlight the strong connection between robustness and privacy. \n \nWe establish information-theoretical lower bounds to show the statistical optimality of our approaches. Technically we use packing lower bounds; however, the novelty of our lower bounds is in capturing the high probability setting.
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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 | 0.013 | 0.048 |
| Meta-epidemiology (narrow) | 0.002 | 0.002 |
| Meta-epidemiology (broad) | 0.003 | 0.002 |
| Bibliometrics | 0.001 | 0.003 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.004 | 0.009 |
| Open science | 0.005 | 0.009 |
| Research integrity | 0.003 | 0.007 |
| Insufficient payload (model declined to judge) | 0.003 | 0.002 |
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".