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CS-675 · computer science · mathematics · Computational statistics

Useful Differential-Privacy Framing

Status Open · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
Standard number systems are $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$. Specialized notation and index ranges follow the cited source and the definitions below.

A differentially private mechanism bounds the change in its output law when one individual's record changes. For neighboring data sets $D,D^{\prime}$, $(\varepsilon,\delta)$-privacy requires $\Pr[M(D)\in E]\leq e^\varepsilon\Pr[M(D^{\prime})\in E]+\delta$ for every event $E$.

Record scope and qualifications.
The catalogue title is retained as metadata. Named certificates, contracts, bounds or frameworks in that title are not asserted as results of the original source extract. The supplied registry flags this extraction as unrecoverable. It is displayed as a literal source fragment; the definitions provide context rather than a replacement formal statement.

Neither the statistical task nor useful accuracy is defined by the source framing.

Source-record limitation: Source flags the statement as unrecoverable: The official report uses this broad sentence as expository framing and does not pose it as a determinate problem separate from the explicit algorithmic question retained below. No separate clean statement is promoted; the original record remains publicly visible and retained for provenance. The source text below is rendered literally where necessary; damaged formulas and composite extracts are not silently repaired into a new mathematical claim. The supplied catalogue marks this record as unrecoverable. Its extract and terminology are documented, but a complete original problem has not been reconstructed.

Statement recorded in the supplied source.
What is the complexity of ensuring differential privacy for useful statistical data analysis? [S2]
Source status.
The supplied catalogue labels this source record open; that metadata was checked against the supplied snapshot on 2026-10-11. The label is preserved for provenance and is not an independent verification of present open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-21.

Short English statement

The fragment asks about private statistical analysis that remains useful.

Sources

  • S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30002236 (OWR-12177-005). Catalogue curation and metadata: CC BY 4.0. The source extract is retained; definitions, English summaries and qualifications are editorial additions. https://huggingface.co/datasets/ulamai/UnsolvedMath.
  • S2 Complexity Theory (2013), Oberwolfach Reports, DOI 10.4171/owr/2012/54, pp. 3267--3304 https://doi.org/10.4171/owr/2012/54.
  • Source-specific Oberwolfach Reports licences, including CC BY-SA 4.0 where applicable, are retained; see THIRD_PARTY_NOTICES.md.
Preparation and review.
The original source and UnsolvedMath contributors retain attribution for the source extract. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, scope notes and formatting. No new human mathematical authorship or independently verified solution is asserted. Permanent source ID 30002236 and code OWR-12177-005 checked against the complete supplied registry and current repository. Distinct upstream records, including repeated questions and broad or damaged extracts, are retained for complete coverage of the source CS category. A repeated record is not a new independent conjecture. Complete mathematical-equivalence and current-literature review has not been performed. Preparation and final review follow contribution prompts C and D, with source gaps disclosed. Structural and rendering checks do not establish a new theorem or open status.

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

CS-675-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

gpt 6 astra ultra attempt on CS-675

Useful Differential-Privacy Framing

Definitions and mathematical statement

Shared notation.
Standard number systems are $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$. Specialized notation and index ranges follow the cited source and the definitions below.

A differentially private mechanism bounds the change in its output law when one individual's record changes. For neighboring data sets $D,D^{\prime}$, $(\varepsilon,\delta)$-privacy requires $\Pr[M(D)\in E]\leq e^\varepsilon\Pr[M(D^{\prime})\in E]+\delta$ for every event $E$.

Record scope and qualifications.
The catalogue title is retained as metadata. Named certificates, contracts, bounds or frameworks in that title are not asserted as results of the original source extract. The supplied registry flags this extraction as unrecoverable. It is displayed as a literal source fragment; the definitions provide context rather than a replacement formal statement.

Neither the statistical task nor useful accuracy is defined by the source framing.

Source-record limitation: Source flags the statement as unrecoverable: The official report uses this broad sentence as expository framing and does not pose it as a determinate problem separate from the explicit algorithmic question retained below. No separate clean statement is promoted; the original record remains publicly visible and retained for provenance. The source text below is rendered literally where necessary; damaged formulas and composite extracts are not silently repaired into a new mathematical claim. The supplied catalogue marks this record as unrecoverable. Its extract and terminology are documented, but a complete original problem has not been reconstructed.

Statement recorded in the supplied source.
What is the complexity of ensuring differential privacy for useful statistical data analysis? [S2]
Source status.
The supplied catalogue labels this source record open; that metadata was checked against the supplied snapshot on 2026-10-11. The label is preserved for provenance and is not an independent verification of present open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-21.

Short English statement

The fragment asks about private statistical analysis that remains useful.

Sources

  • S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30002236 (OWR-12177-005). Catalogue curation and metadata: CC BY 4.0. The source extract is retained; definitions, English summaries and qualifications are editorial additions. https://huggingface.co/datasets/ulamai/UnsolvedMath.
  • S2 Complexity Theory $2013$, Oberwolfach Reports, DOI 10.4171/owr/2012/54, pp. 3267--3304 https://doi.org/10.4171/owr/2012/54.
  • Source-specific Oberwolfach Reports licences, including CC BY-SA 4.0 where applicable, are retained; see THIRD_PARTY_NOTICES.md.
Preparation and review.
The original source and UnsolvedMath contributors retain attribution for the source extract. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, scope notes and formatting. No new human mathematical authorship or independently verified solution is asserted. Permanent source ID 30002236 and code OWR-12177-005 checked against the complete supplied registry and current repository. Distinct upstream records, including repeated questions and broad or damaged extracts, are retained for complete coverage of the source CS category. A repeated record is not a new independent conjecture. Complete mathematical-equivalence and current-literature review has not been performed. Preparation and final review follow contribution prompts C and D, with source gaps disclosed. Structural and rendering checks do not establish a new theorem or open status.

Research attempt

Literature, source context and scope

The full original catalogue body and source qualifications are preserved above. The relevant primary workshop passages were read in the cached publisher report. Historical claims and subsequent attributed results are distinguished from the mathematical work below. A restricted example, standard lemma or known theorem is not counted as new progress on the entire original target. Missing assumptions and repeated or rhetorical extracts are not silently turned into different solved problems.

Postprocessing helps utility only relative to a model

We test a precise way to improve a release without spending further privacy. Suppose $M$ obeys $(\varepsilon,\delta)$-differential privacy, meaning $P_D(A)\le e^\varepsilon P_{D'}(A)+\delta$ for adjacent databases and measurable output events. Let $K$ be a randomized output transformation whose conditional distribution depends only on the released value, not again on the private database.
Proposition: The transformed mechanism obeys the same privacy inequality. For a true empirical mean $\mu\in[0,1]$, clipping a Laplace release $Y=\mu+Z$ to $[0,1]$ weakly decreases squared error on every outcome. With $Z$ of scale $b=1/(n\varepsilon)$, $n\ge1$, $\varepsilon>0$, the clipped mean-square error is at most $2b^2$.
Proof: For any transformed-output event $B$, set $f(y)=K(y,B)\in[0,1]$. The layer-cake identity $f(y)=\int_0^1\mathbf1_{\{f(y)>t\}}dt$, followed by integration of the privacy inequality for these level sets, gives \[ \int f\,dP_D\le e^\varepsilon\int f\,dP_{D'}+\delta. \] This is exactly privacy of the transformed release. If $Y<0$, clipping moves it to 0, which is no farther from any $\mu\in[0,1]$; if $Y>1$, the analogous statement holds at 1; in the interval the value is unchanged. Thus the squared-error comparison is pointwise. Finally, \[ \mathbb E Z^2=b^{-1}\int_0^\infty z^2e^{-z/b}\,dz =b^2\int_0^\infty t^2e^{-t}\,dt=2b^2, \] where two integrations by parts evaluate the last integral. A one-row change changes the empirical mean by at most $1/n$, and the Laplace density ratio is at most $\exp((1/n)/b)=e^\varepsilon$, giving the stated private starting release. ∎
The attempted inference "privacy itself ensures statistical usefulness" fails even for this setting: the constant release 0 is 0-private but has squared error 1 on a database of all ones. Clipping uses the known parameter interval and a particular loss; it is not a universal utility theorem. A transformation that reopens the database is not covered by the kernel argument.

Verification and remaining gaps

Clipping was checked on a rational grid of means and raw outputs, including both boundaries. The Laplace variance bound was evaluated at explicit parameters. The source discusses broadly useful private statistics, not just this bounded scalar mean. No optimal general mechanism, multidimensional query guarantee, or population-risk result is obtained. The worked result is a standard postprocessing lemma and a limited utility improvement, counted as zero original-target progress.

References and source attribution

Primary workshop report: https://doi.org/10.4171/owr/2012/54. Relevant passages read: 2012/54 pp. 3278--3279, private statistical analysis and utility framing.

All retrievals were checked on 11 October 2026. Original question attribution remains in the preserved Sources section. External theorems are attributed and not claimed as new discoveries or fully verified merely by citation.

Integration review.
Codex (GPT-6; exact serving revision not exposed) reviewed the written arguments and expanded layout and source-reference wrappers. Repository metadata and the production web renderer were checked using Python 3.13 and JavaScript with MathJax 3.2.2. This model-assisted review is not independent human proof verification; no PDF compilation or formal proof checking is asserted.

Completion Estimate

COMPLETION: 0% No substantiated new progress toward the entire original target is claimed. The worked results and failed route are delimited above; a known result, illustrative model, or auxiliary obstruction is not counted as original-target completion.

Final status and attribution

Attempt status: unresolved. This describes this research attempt, not a new ruling on the source catalogue's status or on external literature claims. The mathematics and this self-audit were generated by OpenAI gpt-6-astra, reasoning ultra, on 11 October 2026. Codex (GPT-6; exact serving revision not exposed) orchestrated the batch, reviewed the written mathematical arguments, and contributed formatting/repository review. No human mathematical authorship, independent human proof checking, or formal verification is claimed. The historical preparation paragraphs in the reproduced catalogue body concern that body only.

Contribution prompt A was applied to the complete definition and supplied source record, followed by the adversarial checks of prompt B. Local reproduction files are batch019/checks.py, computations.json, and the per-ID audit JSON. They were not public links at preparation time. Numerical checks are finite and do not prove unrestricted results. No LaTeX engine was found locally; compilation was not performed by this research agent.

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