PaperlayerView exact revision on arXiv

Source-linked AI summary

Target-Dependent Limits of Causal Repair: A Leading-Log Frontier in a Gaussian Model

Qinchuan Cheng, Jiaqi Liu, Ruixuan Xie

arXiv:2610.00424v1stat.MLcs.AIcs.LG

TL;DR

The paper asks how much learning accuracy must be traded to assess the realized gain of a trained repair, rather than only a predictor’s potential improvement. In a structured scalar Gaussian experiment, it jointly optimizes learner and assessor and uses diagnostic abstention. It establishes a learning–assessment frontier and transfers its leading limits to adaptive sampling.

  • Problem

    A predictor may be known to be improvable without knowing how much a particular trained repair improves it, and the assessment target depends on the repair’s chosen direction.

    Source: §1 · p. 1

  • Method

    The paper jointly optimizes a repair and its assessor under a uniform learning-error constraint, using diagnostic abstention in a structured Gaussian experiment.

    Source: §1 · pp. 1–2

  • Results

    At learning MSE O(1/k), every feasible repair incurs a k^-2 gain-assessment floor; the leading-log frontier is sharp and its lower bounds transfer to adaptive sampling with one extra cheap observation.

    Source: §1 · pp. 1–2, §3 · p. 3

  • Takeaways & Limitations

    In this experiment, assessing a realized learned gain requires different information from assessing oracle potential, and the frontier should not be read as universal numerical superiority.

    Source: §1 · p. 2, §9 · p. 9

    The attainable frontier relies on identifiable magnitude channels and known geometry; multiplicative risk constants and the exact second-order threshold remain unresolved.

    Source: §1 · p. 2, §7 · p. 8

Abstract

from arXiv · show

Knowing how much a causal predictor could improve need not reveal the gain of the repair actually learned. We quantify this gap in a scalar Gaussian causal experiment with known intervention geometry: auxiliary data identify effect magnitude up to bounded contamination, while diagnostics identify direction. The target is the squared-loss gain of the realized trained repair relative to a fitted reference. Jointly optimizing the learner and assessor under uniform learning MSE $\eta$ avoids the trivial solution of making no repair. At the usual $1/k$ learning scale, every feasible learner incurs a $k^{-2}$ assessment floor, even when oracle potential is estimable at a faster rate. In the magnitude-rich regime, we characterize a sharp leading-log frontier: the assessment exponent is $\min{\ell_k,2k\eta_k/U}$ to first relative order, where $\ell_k=\log(1/(k^2E_k))$ and $E_k$ is auxiliary precision. A diagnostic-abstention rule attains this exponent with unknown nuisance parameters. We also bound the critical allowance window and transfer the frontier to adaptive sampling by exact Gaussian simulation. Finite-grid experiments distinguish sign-tail suppression from total MSE and expose conservative finite-budget behavior. The result isolates how the assessment target changes information requirements in this experiment; it is not a general causal identifiability claim.

1 INTRODUCTION

The paper distinguishes a predictor’s potential improvement from the gain of the particular repair a learner actually trains. In a structured scalar Gaussian experiment, it jointly optimizes learning and assessment to characterize the trade-off and its limits.

1 INTRODUCTION

  • Auxiliary interventions can reveal causal-error magnitude while a small diagnostic sample reveals direction, so assessing a trained repair depends on the direction it actually chose.Oracle improvement depends on magnitude alone; more precise auxiliary data can improve one assessment problem while leaving another limited by diagnostics.Source: §1 · p. 1
  • The target is the realized squared-loss gain of the trained repair relative to a fixed reference, with learner and assessor optimized jointly under a uniform learning-error constraint.Without that constraint, returning the reference makes gain zero and perfectly assessable; fresh evaluation data alone are averaged out.Source: §1 · p. 1
  • In the structured scalar Gaussian experiment, known intervention directions create an effect-dependent variance that separates magnitude from sign.The variance identity is an assumption with a structural realization, not a generic property of causal interventions.Source: §1 · p. 1
  • At MSE O(1/k), every feasible repair has a k^-2 gain-assessment floor; in a magnitude-rich regime, an allowance of order ℓk/k permits assessment at auxiliary precision with threshold constant U/2.A diagnostic-abstention rule attains the leading exponent uniformly over unknown effects, noise, contamination, and references satisfying the stated moment contract.Source: §1 · pp. 1–2
  • The frontier is first-order sharp on a logarithmic risk scale, while multiplicative risk constants and the exact second-order threshold remain unresolved.A parameter-independent Gaussian simulator transfers joint lower bounds to adaptive sampling at the cost of one extra cheap source observation, without ruling out finite-sample constant improvements.Source: §1 · p. 2

2 EXPERIMENT, REFERENCE, AND ASSESSMENT TARGET

The experiment uses a fitted observational reference and separates magnitude information from diagnostic direction information under known Gaussian intervention geometry. It distinguishes assessment of oracle potential, a fixed repair, and the realized gain of a jointly chosen repair.

2 EXPERIMENT, REFERENCE, AND ASSESSMENT TARGET

  • The parameter is q = (c, h, τ, ζ), with |c| ≤ C0, |h| ≤ R, τ ∈ [L, U], and contamination 0 ≤ ζ ≤ ∆ ≤ 1/2; the known contamination bound ∆ does not reveal ζ.The intervention variance is V = τ + h^2 + ζ.Source: §2 · p. 2
  • The data comprise observational pairs for c, up to n cheap N(0,V) responses, and k diagnostic N(xθ,V) responses with |x| ≤ 1.Fixed direct sampling sets every diagnostic direction to x = 1; adaptive policies may choose directions, interleave channels, randomize, and stop subject to pathwise caps.Source: §2 · p. 2
  • Known intervention geometry makes the variance channel identify magnitude up to contamination, while the diagnostic channel supplies direction; these are assumptions of this experiment.Here c is the observational regression coefficient and θ the intervention coefficient, and the confounding correction h also contributes h^2 to interventional variance.Source: §2 · p. 2
  • The observational training rule fixes reference b, while learner f outputs repair coefficient af(X); the realized gain is evaluated on fresh assignments and can be negative.The fitted oracle potential repairs to unknown θ, whereas population-reference gain H = h^2 also replaces b by c.Source: §2 · p. 2
  • The assessment targets distinguish oracle potential H and A from fixed-repair gain Ga and realized learner gain Gf, with the latter jointly constrained by learning MSE ≤ η.The stated decision classes separately optimize the assessor, or learner and assessor together; adaptive versions also optimize the sampling policy.Source: §2 · p. 3

3 A SHARP LEADING-LOG LEARNING–ASSESSMENT FRONTIER

The results give a minimax floor at diagnostic-scale learning accuracy, then characterize how the assessment exponent grows with learning allowance until auxiliary precision limits it. The critical allowance requires a corrected center, and the statements concern leading logarithms rather than exact finite-sample risk.

3 A SHARP LEADING-LOG LEARNING–ASSESSMENT FRONTIER

  • Rfix(C2/k) ≍ Rad(C2/k) ≍ rN + k^-2, and the lower bound holds for every feasible learner while a fixed direct construction attains the upper bound.No fourth-moment constraint is imposed on the learner class.Source: §3 · p. 3
  • For clean m = n = k^3, oracle-potential assessment risk is k^-3, while every repair with 1/k learning accuracy incurs gain-assessment risk at least of order k^-2.This comparison motivates the allowance-dependent frontier.Source: §3 · p. 3
  • When ℓk = o(k), every fixed C > U/2 gives Rk(Cℓk/k) ≍ Ek, while achieving Rk(ηk) = O(Ek) requires lim inf kηk/ℓk ≥ U/2.The leading exponent grows with learning allowance until it saturates at auxiliary precision; Figure 1 depicts Theorem 2, not exact finite-sample risk.Source: §3 · p. 3, Figure 1 · p. 4
  • The critical allowance is centered at zc = Uℓk/(2 − ℓk/k): reaching auxiliary precision requires kηk ≥ zc − O(wk), and Algorithm 1 attains it at zc + Cwin wk.The correction zc − Uℓk/2 can exceed the stated window even when ℓk = o(k); the exact second-order coefficient and attainability at zc remain unsettled.Source: §3 · p. 3

4 DIAGNOSTIC ABSTENTION: CONSTRUCTION AND PROOF MECHANISM

The construction combines magnitude estimation with a diagnostic gate that abstains unless direction is sufficiently supported. Its accepted wrong-sign tail has a sharper exponent, while a joint tail argument controls nuisance effects without a diverging multiplier.

4 DIAGNOSTIC ABSTENTION: CONSTRUCTION AND PROOF MECHANISM

  • Algorithm 1 pools cheap and centered diagnostic energy into Q, then applies a magnitude gate and a diagnostic threshold before either abstaining at (b, 0) or repairing and reporting Q.The declared rule attains the leading exponent and critical-window upper bound with unknown effect, variance, and sign.Source: §4 · p. 4
  • Requiring |M − b| ≳ r moves an accepted wrong-sign crossing from approximately −r to −2r, changing the Gaussian-tail exponent from kr^2/(2V) to 2kr^2/V.Correct-sign rejection is controlled by d(Q), with its cost absorbed by learning slack.Source: §4 · p. 4
  • A joint square-root-tail inequality controls the envelope’s middle term without a nuisance multiplier diverging as ϵ shrinks, yielding the bound for every fixed a0 < 2/U.The first and last terms cost O(Ek).Source: §4 · pp. 4–5
  • Uniform learning accuracy forces nontrivial repair despite residual sign ambiguity, and a posterior square identity gives the lower exponent 2/U without arbitrary learner fourth-moment assumptions.Retaining the signal’s r^2 contribution in the denominator gives ψk(z) = 2z/(U + z/k), whose equality to ℓk yields zc.Source: §4 · p. 5
  • An orthogonal Gaussian reservoir simulates any capped adaptive policy from fixed counts (m, n + 1, k), preserving the learner’s gain and learning error and transferring the lower bounds.The one-sample change perturbs ℓk by o(1), preserving both leading results.Source: §4 · p. 5

5 CHANGING THE TARGET CHANGES THE INFORMATION LIMIT

The assessment rate depends on the target, even with the same data: oracle potential, population gain, and realized learned gain have distinct risks. The learned-gain target retains an additional k^-2 assessment floor.

5 CHANGING THE TARGET CHANGES THE INFORMATION LIMIT

  • Rfix(Ga) ≍ min(EN, k^-1) + k^-2, while the population gain H and fitted oracle potential A have rates min(EN, K^-1) and min(EN, k^-1), respectively.The fixed learned-gain result is for the direct clipped-mean learner; K = min(k, m).Source: §5 · p. 5
  • When diagnostics are plentiful, the fitted target avoids a floor incurred by the population-reference target; when magnitude information is plentiful, the oracle target removes an assessment obstruction.These are strict separations between the stated targets, not identical learner optimizations.Source: §5 · p. 5
  • The transfer bound gives equivalent risks when k^2rN → ∞, but it cannot establish the additional k^-2 lower bound, which requires a sign pair.The bound follows from A − Ga = (θ − a)^2 and the clipped mean’s fourth error moment.Source: §5 · p. 5
  • Table 1 compares different decision classes: its first two columns optimize assessment, while its last fixes both the direct design and clipped-mean learner.Joint learner selection is treated separately in Theorem 1.Source: Table 1 · p. 6

6 NUMERICAL EVIDENCE

Finite-grid experiments examine target-dependent risk scales, wrong-sign tail suppression, and the gap between asymptotic attainment and finite-budget performance. They also show how calibration and conservative variance bounds can limit practical gains.

6 NUMERICAL EVIDENCE

  • The experiments retain seven mechanism and baseline suites plus 1,920 finite-budget and 42 variance-bound configurations; these finite-grid risks are not minimax suprema or uniform-feasibility estimates.Methods within a cell share exact Gaussian sufficient-statistic draws, and conditional Gaussian integration resolves rare diagnostic errors.Source: §6 · p. 6
  • Across k = 16 to 4096, k^3-scaled oracle MSE ranges from 15.38 to 17.57, while k^2-scaled learned-gain MSE ranges from 13.08 to 14.21.The comparison uses clean m = n = k^3, c = 0, h = .3, τ = 2; subtracting bV/k improves the learned-gain constant without removing the scale difference.Source: §6 · p. 6
  • At k = 224 and h^2 = 1.1η, the fixed-gate normalized wrong-sign exponent rises from .141 to 1.003 as z grows, but total MSE stays near 8Ek because calibration dominates.Magnitude-only gating reaches .274, and fast slack reaches .989; Table 2 summarizes seven suites using high-quantile and worst observed MSE ratios.Source: §6 · p. 6, Table 2 · p. 7
  • On the clean magnitude-rich grid, diagnostic gating changes total MSE by less than 4 × 10^-24 relative to matched magnitude gating, while stress-grid learning MSE can exceed the allowance.Projected fallback has a grid-maximum learning ratio of .291 at the displayed allowance, but this numerical maximum is not a proof of uniform feasibility.Source: §6 · p. 7
  • With k = 16384, h^2 = 1.1η, and z = 256, using V* = 4 gives log wrong-sign MSE −130.3, versus −78.6 using 4V*; both total MSEs are 3.75 × 10^-12.A larger valid bound delays tail suppression, which helps total MSE only before calibration dominates.Source: §6 · p. 7

7 SCOPE AND PRACTICAL INTERPRETATION

The generalized lower-bound result extends beyond the quadratic variance identity under stated conditions, but the attainable frontier remains tied to stronger structural assumptions. Practical interpretation is further limited by conservative bounds and by what the learning constraint guarantees.

7 SCOPE AND PRACTICAL INTERPRETATION

  • The k^-2 obstruction survives arbitrary sign-invariant auxiliary observations and unequal channel variances when diagnostic variance is bounded away from zero and infinity and the reference is sufficiently accurate.These are lower bounds, not a matching generalized frontier; for diagnostic variance tending to v0, the sign pair yields a one-sided exponent bound 2/v0.Source: §7 · p. 7
  • The attainable Ek frontier is not asserted when cheap variance is an unknown monotone function of |h|, experimental variances differ, or unknown directions destroy the sign-invariant channel.Under those departures, the magnitude calibration bound, pooled chi-square identity, or parameter-independent reservoir no longer applies as written.Source: §7 · p. 8
  • A larger fixed valid V* preserves the leading exponent but enlarges the diagnostic discount and sufficient-window constants, and bounds growing with k are outside the theorem’s assumptions.The diagnostic threshold is positive only if kQ > 8V*log(8Q/(ϵη)); otherwise the rule is magnitude-only on that draw.Source: §7 · p. 8
  • The uniform learning-MSE constraint controls average excess loss, not positive realized gain or Prq(Gf < 0); objectives requiring positive gain, confidence bounds, or intervention costs need separate analysis.Even positive average gain needs EqA > η.Source: §7 · p. 8

8 RELATION TO PRIOR WORK

The paper distinguishes its realized-gain assessment target from established work on loss estimation, constrained risk, and intervention design. Its contribution is an experiment-specific minimax assessment frontier for a quality-constrained chosen repair, not a general acquisition rule.

Assessing a learned rule

  • The paper’s distinct target is minimax assessment of a jointly selected, quality-constrained repair’s realized gain in a two-channel causal experiment.The distinction between parameter estimation and loss assessment has established precedent, including Cai & Guo (2018) and Bates et al. (2024).Source: §8 · p. 8

Constrained risk, rejection, and nonlinear functionals

  • Joint control of magnitude error, reference error, and accepted wrong-sign tails yields the uniform constant 2/U and the critical window.The supporting strength rates follow a modulus calculation.Source: §8 · p. 8

Intervention design and target-dependent information acquisition

  • Unlike intervention-design work that optimizes what to learn or which action to take, this analysis assesses the chosen learner’s realized gain under a uniform parameter-MSE constraint.The adaptive result compares statistical experiments for that random target; it does not propose a generally optimal acquisition function or rule out adaptive gains in other causal designs.Source: §8 · p. 8

Confounding and Gaussian experiment comparison

  • Known geometry isolates the magnitude–direction information split, and an exact one-extra-sample simulator preserves the learner’s target and second-moment quality constraint.A bounded-loss approximation alone would not transfer the unrestricted learner class; the same-budget connection is to sample amplification.Source: §8 · p. 9

9 CONCLUSION

In the specified Gaussian experiment, learning, realized-gain assessment, and oracle-potential assessment require different information. Diagnostic abstention trades learning accuracy for a larger assessment exponent, while the sharp frontier and adaptive transfer remain limited to the stated setting.

9 CONCLUSION

  • A uniform learning constraint prevents the vacuous zero-gain solution, while diagnostic abstention trades some learning accuracy for a larger assessment exponent.The leading-log frontier is sharp, and the corrected critical allowance is localized within a proved window.Source: §9 · p. 9
  • Exact simulation carries the assessment limits to adaptive designs, but finite-budget comparisons caution against reading them as universal numerical superiority.Source: §9 · p. 9
  • Extending the attainable frontier beyond identifiable magnitude channels and known geometry remains open.Source: §9 · p. 9

REPRODUCIBILITY STATEMENT

The supplement documents the synthetic experiments, their implementation, and checks intended to support reproduction of all nine numerical suites.

REPRODUCIBILITY STATEMENT

  • The reproduction command reruns all nine numerical suites, regenerates every figure and numerical table, and checks the released outputs.Source: REPRODUCIBILITY STATEMENT · p. 10
  • The supplement provides fixed parameter grids, independent-cell seeds, paired Monte Carlo standard errors, machine-readable results, numerical tests, and source/result SHA-256 hashes.Source: REPRODUCIBILITY STATEMENT · p. 10
  • Each suite samples the exact Gaussian sufficient-statistic distribution, subject to floating-point arithmetic; experiments use synthetic data and a CPU.Log-domain conditional integration handles rare diagnostic errors.Source: REPRODUCIBILITY STATEMENT · p. 10

AI USE STATEMENT

Generative AI tools assisted across research framing, mathematical work, experimental work, and manuscript editing. The authors describe verification artifacts while clarifying that these do not constitute formal proof verification.

AI USE STATEMENT

  • Generative AI tools assisted with research framing, formulation and critical review of mathematical claims, proof writing, related-work search, experimental work, and manuscript editing.Source: AI USE STATEMENT · p. 10
  • The assistance included the fast-slack construction, sign-invariant lower-bound extension, citation verification, and executable synthetic experiments and analysis.Source: AI USE STATEMENT · p. 10
  • Analytic derivations, numerical checks, deterministic replay, and compilation checks are reported as verification artifacts, not formal proof verification.Responsibility for submitted claims, code, citations, and disclosure rests with the authors.Source: AI USE STATEMENT · p. 10

ETHICS STATEMENT

The paper states that its statistical assessment guarantees do not certify intervention benefit or safety. Its proof guide covers the structural magnitude-and-direction channels, pooled variance estimation, and supporting empirical implementation.

ETHICS STATEMENT

  • The guarantees concern statistical assessment MSE under explicit assumptions, not whether a realized intervention is beneficial or safe.Gain estimates are not deployment authorization without an appropriate decision criterion and model validation.Source: ETHICS STATEMENT · p. 10

A PROOF AND IMPLEMENTATION GUIDE

  • The proof and implementation guide maps appendices to the lower bounds, leading exponent, critical-window analysis, adaptive transfer, oracle rates, and empirical protocols.It also identifies a fixed clipped-mean learner and a separate contamination fallback.Source: A PROOF AND IMPLEMENTATION GUIDE · p. 12

B.1 A STRUCTURAL MODEL WITH MAGNITUDE AND DIRECTIONAL CHANNELS

  • The structural model uses known intervention geometry and a quadratic variance contribution from the scalar effect to separate magnitude from direction.The construction assumes common experimental measurement noise; without these restrictions, a variance difference need not identify a squared causal correction.Source: §B.1 · p. 12

B.2 POOLED VARIANCE FOR EVERY DIAGNOSTIC BUDGET

  • For every diagnostic budget, the pooled variance estimate has squared error bounded by C_Q E_N, and clipping to [0, R^2] reduces that error.The direct diagnostic mean is independent of the centered residuals; a single diagnostic contributes no residual degree of freedom.Source: §B.2 · p. 13

C WHICH ASSUMPTIONS SUPPORT WHICH CONCLUSIONS?

This section separates conclusions supported by sign-invariant diagnostics from those that require calibrated magnitude and variance assumptions, and clarifies the scope of the joint-risk and leading-log results. It establishes a finite sign-based lower bound, a necessary critical-window allowance, and a joint-tail inequality for gated assessment error under only the stated second-moment reference contract.

C WHICH ASSUMPTIONS SUPPORT WHICH CONCLUSIONS?

  • At learning MSE C2/k, every feasible learner incurs a k^-2 assessment-risk floor even when auxiliary information reveals nothing about effect magnitude.This lower bound applies when m ≥ k and follows from a sign pair with identical auxiliary-transcript laws.Source: C WHICH ASSUMPTIONS SUPPORT WHICH… · p. 13
  • The sign lower bounds extend to capped adaptive diagnostics, but they do not establish auxiliary identifiability or an attaining procedure.For v(r) → v0 > 0 under the stated vanishing-η and reference conditions, the lower bound also yields a leading sign-obstruction exponent; the adaptive extension is pairwise.Source: C WHICH ASSUMPTIONS SUPPORT WHICH… · pp. 13–14
  • The matching upper bound depends jointly on accurate magnitude estimation, a valid known diagnostic-variance bound, and the local variance relation.Monotonicity of an unknown variance function alone does not provide magnitude calibration; different calibrations can agree on observed variance while assigning different magnitudes.Source: C WHICH ASSUMPTIONS SUPPORT WHICH… · p. 14

D JOINT CHOICE UNDER A LEARNING-QUALITY CONSTRAINT

  • Under the learning-quality constraint, the joint assessment lower bound includes both the nuisance-risk term r_N and the k^-2 floor.The pooled fitted-potential assessor supplies an upper bound through the nuisance term and the learner’s squared error.Source: D JOINT CHOICE UNDER A… · p. 15
  • Exact Gaussian simulation transfers the adaptive constrained problem to the fixed-source rate, not to equal same-budget risks or a guarantee against constant-factor adaptive gains.The comparison preserves learning error and realized-target risk; it requires common Gaussian variance, scalar direction, and pathwise caps.Source: D JOINT CHOICE UNDER A… · p. 16

E DIAGNOSTIC ABSTENTION AND THE LEADING ASSESSMENT EXPONENT

  • The leading assessment exponent is min{ℓ_k, 2kη_k/U} to first relative order, attained by diagnostic abstention with unknown nuisance parameters.The construction uses a magnitude threshold and a diagnostic-sign gate; its leading-log guarantee combines with the nuisance lower bound, including when ℓ_k/(kη_k) oscillates.Source: E DIAGNOSTIC ABSTENTION AND THE… · pp. 17–18

F JOINT TAIL STABILITY AND A QUANTITATIVE CRITICAL WINDOW

  • The critical-window subsection begins with z = kη and ℓ = log(1/(k²E)) under the same model, reference contract, and same-transcript target.This part provides the setup but no critical-window finding.Source: F JOINT TAIL STABILITY AND A… · p. 18

F.1 FINITE SIGN LOWER BOUND

  • Every feasible learner and assessor incurs a finite positive lower bound from the wrong-sign event under the stated allowance conditions.The proof bounds the lower-tail event without assuming independence; an empty feasible class has infinite minimax risk.Source: §F.1 · p. 19
  • When z is below the critical value z_c, the necessary allowance bound follows by comparing ψ_k(z) with ψ_k(z_c); outside that range, the bound is automatic.The argument uses the mean value theorem, with ψ′_k eventually at least 1/(2U) on the relevant interval.Source: §F.1 · p. 19

F.2 A FINITE JOINT-TAIL INEQUALITY

  • A joint-tail inequality bounds gated assessment error by (5196C_Q + 2316R^2A_ref)E plus a Gaussian-tail term, using only the second-moment reference contract.The constant multiplying E does not grow as ε decreases; an always-inactive gate alone does not certify learning quality.Source: §F.2 · pp. 19–20

F.3 LEARNING FEASIBILITY AND SUFFICIENT ALLOWANCE

This section develops learning-feasibility conditions and allowance-dependent bounds for assessing a learned repair, distinguishing its risk and acquisition cost from those for oracle potential. It also covers adaptive diagnostics, supporting minimax bounds, robust projected assessment, finite-grid checks, and limitations on the guarantees and simulations.

F.3 LEARNING FEASIBILITY AND SUFFICIENT ALLOWANCE

  • For ℓ = ka with 2/3 < a < 1, the center shift divided by √ℓ log ℓ diverges despite ℓ/k → 0, so ℓ = o(k) alone does not control the allowance window.A sufficiently large auxiliary-budget existence example satisfies the final claim, but polynomial auxiliary budgets do not meet its condition; their leading-order equality remains undecided.Source: §F.3 · p. 20, Figure 3 · p. 21

G EXACT AND APPROXIMATE SIMULATION OF ADAPTIVE DIAGNOSTICS

  • With one extra cheap sample, a parameter-independent fixed Gaussian experiment simulates every capped adaptive policy; for k > 0, zero extra cheap samples cannot simulate this fixed-direct source.This establishes the learner-constrained frontier transfer without changing the assessment target, but non-simulability does not imply strict minimax-risk improvement for a particular target.Source: G EXACT AND APPROXIMATE SIMULATION OF… · pp. 21–22

G.1 SAME-BUDGET APPROXIMATE SIMULATION

  • When d = n + k − 1 ≥ 1, same-budget approximate simulation yields relative risk equivalence along sequences with d Rad → ∞, but gives no matching approximation-error lower bound.The kernel preserves O; the stated risk comparison applies to bounded-interval targets T(q, O).Source: §G.1 · pp. 22–23

H SUPPORTING STRENGTH-ASSESSMENT RESULTS

  • The supporting strength results characterize fixed-direct and adaptive minimax MSEs, with constants depending on fixed model bounds and p, and construct an estimator through a model-specific modulus.The modulus yields three regimes by maximizing a functional gap under a distance constraint; the residual upper-bound proof likewise controls risk through that modulus.Source: H SUPPORTING STRENGTH-ASSESSMENT RESULTS · p. 23, §I.2 · p. 24

I.1 INFORMATION AND TESTING TOOLS / I.3 RESIDUAL LOWER BOUNDS

  • When KL is at most 1/8, midpoint testing gives maximum MSE at least 3g²/32 for two targets separated by g.The residual lower bounds combine with the nonzero compensated pair to give F(E, min(k, m)).Source: §I.1 · p. 24, §I.3 · p. 25

I.4 TOTAL STRENGTH

  • For total strength, the diagnostic estimator has a 1/k risk upper bound, and testing gives a matching order lower bound.The lower-bound construction chooses opposite-sign strengths so cheap and observational laws coincide while diagnostic KL stays bounded.Source: §I.4 · p. 25

I.5 SQUARED-LOSS GAINS

  • For population squared-loss gains, the minimax risk is min(E, 1/K) up to constants, combining auxiliary-energy and diagnostic-mean components.The fitted-target upper bound controls the reference-fit error by O(1/m) and the diagnostic estimate by O(1/k), with count selection giving the minimum.Source: §I.5 · p. 25
  • For bounded references, coefficient accuracy is sufficient but not always necessary: when k≥E^-1, diagnostic bounds and lower bounds match at 1/k, while boundedness alone does not ensure that rate for every budget.The zero reference has risk ≍min(1,1/k); at k=0 with Δ=0 and n=m→∞, an inaccurate bounded reference can have constant risk while an accurate one has order 1/m.Source: §I.5 · p. 26

J CONDITIONAL GAIN OF A LEARNED REPAIR

  • Learned-gain assessment has a minimax 1/k rate when kE≥1 and k→∞, and a k^-2 floor on one sequence where oracle-potential assessment is k^-3.Along Δ=0 and n=k=m², learned-gain assessment is m^-2, compared with m^-1 for the population-reference target.Source: J CONDITIONAL GAIN OF A LEARNED REPAIR · pp. 26–27

K ACQUISITION COST FOR THE THREE ASSESSMENT TARGETS

  • For learned-gain assessment, the variance-sampling route requires k≳ϵ^-1/2 and, when λ≥1, the acquisition cost is of order (w+1)/ϵ+λ/√ϵ.These are budget costs for assessing specified targets, not for optimizing the repair learner; the clean-model restriction is essential and the formulas are order statements.Source: K ACQUISITION COST FOR THE THREE… · p. 28

L.3 PROOF OF THEOREM 10

  • For fixed β>2, achieving risk O(k^-β) requires η_k=Ω(log(k)/k), and an explicit allowance of order log(k)/k attains that risk; this is an order threshold, not a sharp constant or complete frontier.At fixed η=C2/k, the same sequence instead has risk of order k^-2.Source: L A LOGARITHMIC ALLOWANCE FOR EASIER… · p. 28, §L.3 · p. 30

L.1 SELECTIVE UPPER CONSTRUCTION

  • The selective construction uses the diagnostic mean and estimated gain magnitude to abstain or repair without an oracle magnitude; an allowance-dependent sign-pair lower bound shows why insufficient learning accuracy limits assessment.The upper bound is uniform over the parameter class; the lower-bound pair is chosen from the declared allowance and sample counts, not observed data.Source: §L.1 · pp. 28–29, §L.2 · p. 29

L.4 INFORMATION-RATIO THRESHOLD AND ADAPTIVE EXTENSION

  • If k^2E_k→0 and ℓ_k=o(k), risk O(E_k) requires η_k=Ω(ℓ_k/k), and selective assessment attains it at η_k∼48V*ℓ_k/k, also under adaptive sampling with pathwise caps.More generally, the results match logarithmic risk orders for intermediate allowances; vanishing allowance permits risk O(E_k) if and only if ℓ_k=o(k). These are order results, not multiplicative risk equivalence.Source: §L.4 · pp. 30–31

L.5 RELATION TO ASSESSING THE LEARNED LOSS

  • When learned-gain risk R_G/E_N diverges, assessing the learned loss has asymptotically the same minimax risk; at R_G=O(E_N), the transfer gives only an upper bound.This target-transfer result is fixed-design only and does not provide the fitted-potential estimator to an arbitrary adaptive transcript.Source: §L.5 · p. 31

L.6 FIXED CONTAMINATION CAN MAKE MORE DATA HARMFUL FOR THIS RULE

  • Under fixed contamination γ, the selective rule can become harmful as data increase: E[a²]→γ, E[G]→−γ, E[(g−G)²]→4γ², and Pr(G<0)→1.This is shown at a legal point with c=h=θ=0 and τ=(L+U)/2.Source: §L.6 · p. 31
  • The fixed-contamination example is a failure of this construction, not an impossibility result: the direct mean has learning MSE (τ + γ)/k but falls outside the rich-information theorem.The passage attributes that exclusion to E_k ≥ γ^2.Source: §L.6 · p. 32

M ROBUST LEARNING AND PROJECTED ASSESSMENT

  • The learner satisfies the stated uniform learning-MSE bound η, while projected assessment is O(k^-3) for clean m = n = k^3 and O(log(k)/k) under fixed contamination.Neither rule takes the subclass bound Δ as input; the k^-3 confidence remainder precludes arbitrary finer precision, and no positive gain is guaranteed.Source: M ROBUST LEARNING AND PROJECTED… · pp. 32–33

N.1 DESIGN, PROVENANCE, AND WHAT IS ESTIMATED

  • Monte Carlo standard errors quantify finite-simulation uncertainty, not uncertainty over a minimax supremum; displayed grids are not exhaustive, and observed zeros do not establish zero population risk.No multiplicity-adjusted significance claims are made.Source: §N.1 · p. 33

N.2 TARGET-SCALE AND FIXED-LEARNER CHECKS

  • The target-scale suite compares Q and Q − bV/k as reports of A and G_a for the same clipped diagnostic-mean learner; the correction changes the assessment constant, not the learner or target.Table 4 also includes a Stein comparator; the supplied passages provide no cell values.Source: §N.2 · p. 34, Table 4 · p. 34

N.3 CONDITIONAL INTEGRATION OF RARE DIAGNOSTIC ERRORS

  • Conditional integration removes diagnostic-mean Monte Carlo variability and retains wrong-sign tail contributions in log arithmetic, but large-z total-risk calculations remain dominated by the nuisance term.The resulting standard errors arise only from nuisance draws; this is not a population tail bound or an importance-sampling confidence interval.Source: §N.3 · p. 35, Table 5 · p. 35

N.4 STRONG ASSESSMENT AND PROJECTION BASELINES

  • 21 of 27 cells favored the joint estimator over the mean method, and 26 of 27 over the variance method; these are descriptive counts, not uniform dominance claims.The residual-strength suite compares all procedures on the same target and data.Source: §N.4 · p. 36, Table 6 · p. 36
  • The rate-selected assessor had lower empirical MSE than the clipped unbiased method in only 2 of 24 cells; its count rule attains risk orders, not optimized finite-sample constants.The unbiased comparator is shown both raw and clipped to the valid target interval.Source: §N.4 · p. 37, Table 7 · p. 37
  • At (c, h, k) = (.3, 0, 16), actual gain was negative in 78.345% of draws; at (.9, .9, 16), 31.8% of repairs hit the clipping boundary and Stein had higher MSE.Both reports assess the same clipped-mean learner; unbiasedness over training data does not certify the observed repair.Source: §N.4 · p. 39, Table 9 · p. 39

N.5 CONTAMINATION, FALLBACK, AND REPORT PROJECTION

The appendices test robustness and finite-budget procedures while preserving each pipeline’s own learner and assessment target. The stress results show that specified procedures can exceed learning allowances outside the asymptotic regime.

N.5 CONTAMINATION, FALLBACK, AND REPORT PROJECTION

  • Fallback changes the learner and therefore the assessment target; raw and projected fallback reports share that target, while the Stein comparator assesses a different learner.Comparisons across these learner pipelines are not same-target assessor comparisons.Source: §N.5 · p. 40, Table 11 · p. 41

N.6 FINITE-BUDGET STRESS AND VARIANCE-BOUND SENSITIVITY

  • At the larger allowance, maximum learning ratios are 10.73 for the magnitude gate and 3.27 for the diagnostic gate; projected fallback gives .291 and meets kη ≥4V∗.These stress results are outside the magnitude-rich asymptotic assumptions and do not locate the minimax threshold.Source: §N.6 · p. 43
  • At z = 16 and 64, no eligible positive diagnostic threshold appears for any bound multiplier; at z = 256, a factor-four bound makes the diagnostic gate coincide with the magnitude rule in nuisance draws.These are observed draw fractions, not claims of zero population probability; total MSE may be indistinguishable even when log wrong-sign MSE changes substantially.Source: §N.6 · p. 43, Table 12 · p. 44

N.7 EXECUTABLE REPRODUCTION AND NUMERICAL CHECKS

  • The numerical checks cover implementation identities and simulator behavior, but the mathematical guarantees rely on proofs; byte-identical replay is not assured across library or platform versions.Exact byte replay is checked within the recorded environment.Source: §N.7 · p. 44
Loading 2610.00424v1…