ScoreQuant

Open question · OPEN-SCORE-PERTURBATION

How robust is the rule to estimated scores?

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Meaning

Relate score estimation error to changes in cell moments, boundaries, and retained information.

Scientific context

There is no perturbation theory bounding the change in cell moments, objective value and cell boundaries from a norm bound on the score estimation error. This absence, together with the missing calibration-to-information propagation (OPEN-CLASSIFIER-CALIBRATION-FI), is why estimated-score provenance never claims exact Fisher semantics anywhere in the library — the downgrade reflects the absence of a theorem, not excess caution.

Statement

First-order reporting and rule-transfer bounds for frozen score rules are audited with corrections (SCORE-ERROR-RETENTION-BUDGET, SCORE-ERROR-RULE-TRANSFER; AUDIT-SCORE-ERROR-BUDGET-001). Remaining: uniform error control plus optimization-gap certification, D_s, refitted rules and sharp second-order constants. No uniform relative/log reporting bound follows from an L2 score-error upper bound alone without retention conditioning.

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Related settled results

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Local graph
Local map of Perturbation theory for estimated scores4 prerequisites to the left, 0 results it enables or questions it raises to the right, 0 boundary cases below.True retained Fisher information under a score proxyTrue retained Fisher…Representation loss and hard-quantization loss separateRepresentation loss a…Reported versus true retention of a frozen label map under a score error: an exact two-term expansion in the Fisher-whitened score errorReported versus true…Applying a frozen rule to the proxy instead of the truth: mislabel mass under a margin condition and an exact Loewner sandwich on the retained informationApplying a frozen rul…Perturbation theory for estimated scoresPerturbation theory f…
Prerequisites to the left, what it enables or raises to the right, boundary cases below. Open in the map.

The question in full

Perturbation theory for estimated scores

Assume ∥s^−s∥L2≤ε\|\hat s-s\|_{L^2}\le\varepsilon or a stronger bound. Control cell moments, IqI_q, the D/DsD_s objective, efficiency, and geometric boundaries under a margin condition. Settled at first order for a frozen rule (closure step 3, 8 Sep 2026, O8): the reporting gap is an exact two-term expansion — alignment ∝d−tr⁡R\propto\sqrt{d-\operatorname{tr}R}, spurious information εZ2/ρmin⁡\varepsilon_Z^2/\rho_{\min} — with the numerator first-order constant attained and no uniform relative/log bound from an upper bound on ε\varepsilon alone (SCORE-ERROR-RETENTION-BUDGET); the boundary displacement is a Markov-plus-margin mislabel bound with an exact Loewner sandwich (SCORE-ERROR-RULE-TRANSFER). Remaining: uniform-over-rules error control plus a proxy optimization-gap certificate (uniform control alone cannot make an exchange-stable solution globally optimal), DsD_s, refitted rules, sharp second-order constants. Target claim: OPEN-SCORE-PERTURBATION.

From OPEN_PROBLEMS.md