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Pragmatic DML with AI-Learned Representations

Andres Aradillas Fernandez, Victor Chernozhukov, Carlos Cinelli, Sven Klaassen, Whitney Newey, Martin Spindler, Jan Teichert-Kluge, Suhas Vijaykumar

arXiv:2610.01935v1econ.EMstat.ML

TL;DR

The paper asks when DML inference using learned representations as controls can be interpreted for the population target. It connects representation errors to target distortion and develops DML inference; in the demand application, the aggregate response remains negative across the reported sensitivity grid.

  • Problem

    The paper asks when DML inference using learned representations as controls can be interpreted for the population target.

    Source: §1 · pp. 1–2

  • Method

    For a broad class of targets, the paper links target distortion to the interaction of outcome-regression and balancing-weight representation errors and develops DML inference for representation-based targets.

    Source: §1 · p. 3

  • Results

    The aggregate implies a negative inverse-rank demand response, and its sign remains negative across the reported sensitivity grid.

    Source: §6 · p. 38

  • Takeaways & Limitations

    The omitted-information identity characterizes how a representation-based target differs from the full-information target.

    Source: §6 · p. 37

    Even when target adaptivity holds, the representation-based score variance may exceed the semiparametric efficiency bound for the full-information target.

    Source: §1 · p. 3

Abstract

from arXiv · show

Text, images, and other rich covariates are increasingly compressed into AI-learned representations and then used as controls in causal analysis. We study when this approach is valid and develop a practical framework for causal inference with learned representations. For a broad class of estimands, an imperfect representation distorts the target causal parameter by the product of two representation errors: one in the outcome regression and one in the balancing weight (or Riesz representer). This yields three constructive results. First, cross-fitted double machine learning (DML) provides valid Wald inference for the representation-dependent target. When representation errors are small, the same interval covers the causal parameter, and it can even attain the semiparametric efficiency bound. Second, fold-wise representation learning (or fine-tuning) is compatible with DML inference for the causal parameter. To this end, we develop convex- and star-aggregation pipelines for learning and combining representations. Third, when representation errors are substantial, we can provide interpretable sensitivity regions and root-$n$ inference for their endpoints. In a multi-modal demand application, seven representation-specific estimates and their star aggregate all imply a negative near-unit elasticity for rank-based price response, and the result remains robust over the reported sensitivity grid.

1. INTRODUCTION

The introduction develops a target-aware approach to causal analysis with learned representations, explaining how representation-based adjustment changes the estimand and characterizing the resulting gap through two representation errors. It combines familiar DML inference with adaptive, fold-wise learning and sensitivity analysis.

1. INTRODUCTION

  • Using (D, Xφ) centers DML on the representation-based average predictive effect, which need not equal the full-information ATE.The central question is when inference for this representation-based target can also be interpreted as inference for the full-information target.Source: §1 · p. 2
  • For continuous linear regression functionals, including the ATE, the target gap is a product of representation errors in the outcome regression and the balancing weight.The representation need only preserve the components relevant to the estimand, rather than reconstructing the full covariate object globally.Source: §1 · p. 2
  • Cross-fitted DML gives inference for the representation-based target, and under target adaptivity its usual interval also covers the full-information target.Target adaptivity requires √n|θ⋆−θφ| = o(1); semiparametric efficiency additionally requires the representation-based regression and representer to approach their full-information counterparts.Source: §1 · p. 3
  • When target adaptivity is not defensible, restrictions on the magnitude and alignment of representation loss yield sensitivity bounds for the full-information target and DML inference for their endpoints.The paper presents this as an alternative to interpreting a single estimate as inference for the full-information parameter.Source: §1 · pp. 3–4
  • Treating representation learning as first-stage learning permits fold-wise tuning without differentiability or stability assumptions on the representation parameter.Inference instead relies on Neyman-orthogonal scores and suitable nuisance prediction rates; fixed embeddings are included as a special case.Source: §1 · p. 3
  • Proper learning over star-shaped approximations and improper star aggregation offer complementary routes to first-stage rates compatible with DML product-rate requirements.Aggregation weights can be fit to pooled out-of-fold predictions rather than within each training fold, with an added remainder-term factor.Source: §1 · p. 4
  • The framework supports comparing, aggregating, and assessing the sensitivity of alternative representations.Source: §1 · p. 4
  • The paper focuses on omitted-information identities, adaptive inference, and sensitivity analysis for learned representations, enabling target-aware embedding choice and evaluation with familiar DML inference.Source: §1 · p. 4

2. ADAPTIVE INFERENCE WITH LEARNED REPRESENTATIONS

This section develops adaptive DML inference with learned representations, distinguishing the representation-defined target from the full-information target and identifying conditions for valid and efficient inference for the latter. It also examines fold-wise representation learning and tuning, including how cross-fitted DML supports inference for the target induced by the learned-representation pipeline.

2. ADAPTIVE INFERENCE WITH LEARNED REPRESENTATIONS

  • Target adaptivity and adaptive efficiency are the conditions under which representation-based DML output can be interpreted as valid and efficient inference for the full-information target.Source: §2 · p. 5
  • When representation tuning is confined to training folds, first-order DML theory applies to the population target defined by the complete tuning and estimation procedure.Source: §2 · p. 5

2.1.1. Setup: full covariates and learned representations.

  • The setup contrasts the full-information target using (D,Z) with the representation-based target using (D,Xφ), which can be treated as fixed when the embedding is frozen or independently trained.For the binary-treatment example, the full-information target is the ATE and the representation-based target is the APE.Source: §2.1.1 · pp. 5–6

2.1.2. Riesz representers and an omitted-information identity.

  • The omitted-information identity shows that target distortion is driven by the product of outcome-regression and balancing-weight representation errors; preserving either component preserves the target.Its bound also allows substantial compression when both components are only partially preserved.Source: §2.1.2 · p. 7

2.1.3. Debiased scores and K -fold cross-fitted DML.

  • The short-target score is Neyman-orthogonal, and its mean drift is negligible at the root-n scale when nuisance-estimation errors satisfy the usual product-rate condition.The cross-fitted DML estimator averages the fold-specific scores.Source: §2.1.3 · pp. 7–8
  • Cross-fitted DML yields asymptotically linear, valid Wald inference and one- and two-sided tests for the representation-specific target θφn.The result concerns the effect defined by the information retained in Xφn, without yet linking it to the full-information target.Source: §2.1.3 · pp. 9–10

2.2. Omitted information bias, target adaptivity, and adaptive efficiency.

  • When the representation target gap is negligible at the √n scale, its Wald interval covers θ⋆; adaptive efficiency additionally yields first-order equivalence to the full-information efficient influence function.The target gap is the product of outcome-regression and Riesz-representer approximation errors.Source: §2.2 · pp. 10–12

2.3. Inference with fine-tuned representations.

  • Fold-wise fine-tuning fits the representation and nuisance functions on training data, then evaluates held-out scores, preserving DML independence without requiring an expansion of the tuning map.First-order inference is governed by prediction errors of the fitted regression and Riesz representer.Source: §2.3 · p. 12

2.3.1. A joint-class view of representation learning.

  • The joint-class pipeline projects outcome and Riesz nuisances onto a common information space, preserving score orthogonality; score drift and the pipeline target gap share mixed-bias structure, second-order when either nuisance is accurate.The outcome and Riesz projections are taken in L2(P) onto the same space.Source: §2.3.1 · pp. 13–14
  • The joint-class view distinguishes the induced target θ0 from the full-information target θ⋆.Source: §2.3.1 · p. 14

2.4. Subordinated tuning and information loss.

  • Profiling quadratic prediction and Riesz losses removes nuisance functions, leaving L2(P) information-loss quantities tied to the representation-dependent target gap.The target gap is governed by the product of outcome-regression and representer errors.Source: §2.4 · pp. 14–15
  • Outcome-side tuning selects the representation that best preserves the full-information conditional mean of Y given D in L2(P).Source: §2.4 · p. 14
  • Representer-side tuning minimizes the representation-based representer’s approximation error to the full-information representer.The restricted representer is an L2(P) projection in the stated setup.Source: §2.4 · pp. 14–15

2.5. DML inference after foldwise tuning.

  • Foldwise inference requires fitted nuisances to converge to projections on a common space, but not to solve profiled tuning problems or use the same intermediate representation.Cross-fitting fixes the fitted functions conditional on training folds, allowing the DML empirical-process arguments to apply.Source: §2.5 · p. 15
  • Cross-fitted DML gives valid confidence intervals for the foldwise pipeline target without requiring tuning-parameter stability or differentiability.The result extends DML to fold-specific learned representations and holds when the stated regularity conditions are met.Source: §2.5 · p. 16

3. LEARNING THE PIPELINE TARGETS OVER H

This section develops learning and aggregation methods for nuisance functions and representations over pipeline candidate classes, using prediction-error bounds to support the DML product-rate requirement. It covers proper, star, and convex aggregation, including pooled aggregation of fold-specific candidates, with guarantees that support DML inference.

3. LEARNING THE PIPELINE TARGETS OVER H

  • For an end-to-end pipeline, the relevant fitted objects are the resulting predictions, and the product-rate requirement can be reduced to learning bounds for two quadratic objectives.The two objectives are for the outcome regression and the Riesz representer.Source: §3 · p. 17

3.1. From quadratic objectives to L2(P) rates.

  • On the shared space H, quadratic excess risks equal squared L2(P) errors, so nuisance learning bounds can verify DML’s product-rate condition.Each nuisance error of order oP(n^-1/4) suffices, and faster learning of one can compensate for slower learning of the other; pipeline approximation and complexity are governed by H.Source: §3.1 · p. 18

3.2. Proper learning on controlled approximations to H .

  • Proper learning uses foldwise empirical risk minimization over controlled approximation spaces, with critical-radius complexity bounds and approximation control to verify nuisance rates.For the representer, the cited rate result assumes bounded-class conditions, including mean-square continuity of the moment function and bounded envelopes for deviation sets.Source: §3.2 · pp. 19–20
  • A finite-dictionary workflow first constructs bounded candidate predictors on one training-fold split, then uses the remaining observations for aggregation.This example matches workflows that try a handful of representations and heads.Source: §3.2 · p. 20
  • Proper learning uses a B-controlled convex ℓ1-hull approximation to span(F), with bounded-assumption ERM and Riesz learning rates linked through the product condition.The resulting practical constraint links the (M,B) choices for the regression and representer learners.Source: §3.2 · pp. 20–21

3.3. Improper learning by star aggregation.

  • Star aggregation turns a finite candidate menu into a mixed predictor in the same information space, with a log(M)/n oracle remainder and potentially closer population projections than any single candidate.The guarantees apply to both regression and Riesz losses; finite-menu bounds require the menu to be independent of the aggregation sample.Source: §3.3 · pp. 21, 23

3.4. Aggregation across folds.

  • Pooled out-of-fold predictions let practitioners fit one set of star weights without retraining candidates on separate aggregation splits, but the weights depend on every fold, including the evaluated fold.Thus cross-fitting holds for the candidates but not for the aggregate; the subsection characterizes this breach as cheap.Source: §3.4 · p. 24

3.4. Aggregation across folds

  • Pooled star aggregation satisfies oracle inequalities against the best single recipe and requires no independence between the candidate menu and the aggregation data.The proof relies on each fold’s candidates being independent of that fold’s observations.Source: §3.4 · p. 25
  • Under Assumption 3, pooled star estimators satisfy the DML expansion when the stated product-rate and uniform empirical-process conditions hold.The empirical-process condition controls the remainder uniformly over star segments; with the stated rates, it reduces to logM = o(sqrt(n)).Source: §3.4 · p. 26
  • Pooled convex aggregation has oracle inequalities and yields the same DML inference conclusions under its stated assumptions and rate conditions.Its estimation term grows like M log(n)/n rather than log(M)/n.Source: §3.4 · pp. 25, 27
  • With the stated nuisance rates, convex aggregation requires M log(n) = o(sqrt(n)); pooling adds no further cost for the weights and can improve projection approximation over any single recipe.The convex combination’s approximation errors are no larger than those of the individual recipes.Source: §3.4 · p. 27

4. SENSITIVITY INFERENCE FOR FULL-INFORMATION TARGETS

The section turns representation loss into interpretable sensitivity intervals for full-information targets, then develops cross-fitted DML inference for their endpoints.

4.1. Identified regions from representation loss.

  • θ−θs = E[(g−gs)(α−αs)], so target distortion is the product of omitted outcome and balancing components.Source: §4.1 · p. 28
  • The sensitivity parameters (ρ,CY,CD) summarize omitted-component alignment and projection-loss magnitudes; user-chosen bounds yield a feasible interval for θ⋆.Source: §4.1 · p. 28
  • The interval is centered at θ0, the population end-to-end pipeline target; with a fixed embedding and containment, its center reduces to θφ.Source: §4.1 · p. 28

4.2. DML inference for sensitivity bounds in the fine-tuned case.

  • Rate conditions apply to end-to-end fitted nuisances, allowing fold-specific representations; cross-fitted DML supports feasible root-n inference for the sensitivity endpoints.Source: §4.2 · pp. 30–31
  • Endpoint Wald inference requires nonzero first-order sampling variation, and the resulting confidence envelope is valid for θ⋆ when maintained restrictions place it in the identified region.Source: §4.2 · p. 31

5. MULTIMODAL DEMAND WITH AI-LEARNED REPRESENTATIONS

The application studies first-difference price responses across seven product-information sets, with estimates showing a negative near-unit response across modalities and a pooled star-aggregation estimate. It also examines sensitivity to omitted-information strength and the empirical importance of learning the two nuisance functions separately, since outcome-prediction quality alone does not capture target preservation.

5.1. A continuous-treatment first-difference specification.

  • First differencing removes time-invariant product effects, and price changes have a causal average-response interpretation if they are conditionally mean-independent of unobserved outcome changes given controls.The outcome is the change in the oriented log inverse-sales-rank proxy and treatment is the change in log price; the response is elasticity-like, not a conventional unit-demand elasticity without further mapping.Source: §5.1 · pp. 32–33

5.2. Outcome and Riesz learners.

  • Outcome and Riesz learners are fitted independently, with candidate selection and aggregation weights learned on training data and estimates evaluated on held-out observations.Fully cross-fitting repeats this construction across folds and averages held-out scores, using each observation for evaluation once.Source: §5.2 · pp. 33–34

5.3. Estimates and star aggregation.

  • All seven modality-specific estimates are negative, with 95% intervals strictly below −1; estimates range from −1.098 for text plus image to −1.034 for tabular data.The full multimodal estimate is −1.069.Source: §5.3 · p. 34, Table 1 · p. 34
  • Star aggregation selects different centers for the outcome and Riesz learners, and its estimate targets the pipeline parameter associated with their shared candidate span under the stated conditions.The image-plus-tabular outcome learner is the center, while the Riesz learner combines a text-plus-tabular center with the full multimodal learner.Source: §5.3 · pp. 34–35
  • The star aggregate estimates a negative near-unit response of inverse-rank demand to price (−1.053; 95% CI [−1.086, −1.021]).A 10% price increase corresponds locally to roughly a 10.5% decline in the inverse-rank demand index.Source: §5.3 · p. 35

5.4. Sensitivity to omitted multimodal information.

  • Removing image and tabular controls changes validation R2 from 0.0314 to 0.0297 and validation Riesz loss from −254.24 to −252.02.The deletion exercise calibrates sensitivity parameters against withholding observed modalities; the reported residual alignment is ρ̂ = 0.8654.Source: §5.4 · p. 35
  • Every modality-specific 95% interval, star-aggregate interval, and full-information sensitivity interval lies below zero; under perfect alignment, the strong-scenario upper endpoint is −0.723 (90th-percentile limit −0.705).The result supports a negative rank-based price response under the maintained differenced design, while the sensitivity region quantifies uncertainty about its economic magnitude.Source: §5.4 · pp. 36–37, Table 3 · p. 36
  • With ρ = 1, the sensitivity interval widens from [−1.137,−1.001] in the mild scenario to [−1.415,−0.723] in the strong scenario.The grid indicates that the sign is highly robust, but the magnitude responds to the assumed strength of omitted information.Source: §5.4 · pp. 36–37, Table 3 · p. 36

5.5. Empirical lessons.

  • Star aggregation selects image-plus-tabular alone for g, but combines text-plus-tabular with the full multimodal candidate for α, illustrating why the nuisances should be learned separately.Image and tabular information barely change outcome R2 but more visibly change Riesz loss, so prediction quality alone does not summarize target preservation.Source: §5.5 · p. 37
  • Under the maintained first-difference design, the negative response is quantitatively stable across the observed representation menu and sensitivity grid; train-validation inference is honest under stated sampling conditions.The richest multimodal pipeline is an observed-information benchmark, and the sensitivity scenarios extend the analysis toward the full-information target.Source: §5.5 · p. 37

6. CONCLUSION

The paper develops target-aware causal inference with learned representations, linking target distortion to discarded outcome-regression and Riesz-representer components. It combines DML inference, representation aggregation, and sensitivity analysis, and the demand application finds a negative response across representations and the reported sensitivity grid.

6. CONCLUSION

  • For Riesz-linear functionals, representation replacement changes the target through one mixed-bias term: the inner product of discarded outcome-regression and Riesz-representer components.The identity focuses attention on information relevant to the estimand rather than global reconstruction of the original covariates.Source: §6 · p. 37
  • Cross-fitted DML provides valid inference for the representation-specific target; target adaptivity and adaptive efficiency can make inference valid for the full-information target and first-order equivalent to its efficient influence-function inference.For representations tuned on the sample, fold-wise tuning is absorbed into nuisance learners, making inference depend on prediction rates rather than a linear expansion of the tuning algorithm.Source: §6 · pp. 37–38
  • Convex aggregation suits explicit sieves and convex classes, while star aggregation offers a computationally simple, statistically sharp approach for large nonconvex menus of prompts, embeddings, and heads.Both schemes can fit weights to pooled out-of-fold predictions, avoiding nested sample splitting.Source: §6 · p. 38
  • Sensitivity analysis translates restrictions on representation loss into an identified region for the full-information target, with inference for its endpoints and the region as a whole.The demand application uses the region to quantify uncertainty about omitted information.Source: §6 · p. 38
  • All seven representation-specific estimates and the star aggregate imply a negative inverse-rank demand response to price, with the sign remaining negative across the reported sensitivity grid.The application’s intervals widen to show how uncertainty changes under sensitivity analysis.Source: §6 · p. 38

APPENDIX A. PROOFS FOR SECTION 2

Appendix A collects proofs for Section 2 while keeping the main text focused on applied interpretation.

APPENDIX A. PROOFS FOR SECTION 2

  • Appendix A collects the proofs for Section 2.The appendix states that this organization keeps the main text focused on applied interpretation.Source: Appendix A · p. 41
  • The appendix uses somewhat more formal notation than the main text.It includes empirical-process notation.Source: Appendix A · p. 41
  • The appendix’s notation follows the standard style of.Source: Appendix A · p. 41

A.1. Empirical process notation for proofs.

This subsection introduces empirical-process notation used in the proofs and states that the symbols are confined to the appendix.

A.1. Empirical process notation for proofs.

  • For a measurable function f: O → R, the notation is introduced under the condition P f < ∞.Source: §A.1 · p. 41
  • For each fold I_k, the appendix denotes its size by n_k := |I_k|.Source: §A.1 · p. 41
  • The appendix states that these symbols are used only in the appendix.Source: §A.1 · p. 41

A.2. Proof of Proposition 1.

The proof of Proposition 1 establishes the target identity by showing that discarded components are orthogonal to functions of the representation.

A.2. Proof of Proposition 1.

  • The proof identifies the long and short targets as θ⋆ = E[g⋆α⋆] and θφ = E[gφαφ].Source: §A.2 · p. 41
  • Conditioning the outcome component on Wφ makes its discarded residual orthogonal to square-integrable functions of Wφ, including αφ.Source: §A.2 · p. 41
  • Compatibility makes the short functional the restriction of the long functional to Wφ-measurable functions, yielding αφ(Wφ) = E[α⋆(W) | Wφ].The proof then uses orthogonality of the discarded representer component to establish equation (6), with equation (7) following from Cauchy–Schwarz.Source: §A.2 · p. 41

A.3. Proof of Theorem 1.

The proof establishes the cross-fitted estimator’s expansion by controlling its bias and empirical-process remainders, then derives asymptotic normality and variance consistency. It also proves the approximation and variance claims of Lemma 1.

A.3. Proof of Theorem 1.

  • The cross-fitted estimator’s bias remainder is second order: Riesz representation cancels terms in expectation, and the rate condition makes the remainder negligible.The proof also shows the empirical-process remainder is negligible under L2 consistency.Source: §A.3 · p. 42
  • The estimator has an asymptotic normal limit after the negligible remainders are removed, using a central limit theorem and Slutsky’s theorem.The proof treats fixed and drifting representations, using Lyapunov’s theorem for the triangular-array case.Source: §A.3 · pp. 43–44
  • The cross-fitted variance estimator is consistent by comparing fitted and oracle scores and controlling their difference across folds.The proof uses conditional arguments and uniform integrability for a drifting representation.Source: §A.3 · pp. 44–45

A.4. Proof of Lemma 1.

  • For Lemma 1, condition (20) makes the specified approximation terms vanish, while L2 convergence yields variance convergence.The proof uses an algebraic expansion and the triangle inequality.Source: §A.4 · p. 45

A.5. Proof of Theorem 2.

These proofs establish efficiency under target adaptivity, characterize restricted regression and Riesz objects through projections, and extend the inference argument to the restricted target. They also provide proof details for Section 3, including finite-class aggregation bounds.

A.5. Proof of Theorem 2.

  • Under target adaptivity, the estimator is centered at the full-information target and attains asymptotic efficiency.The proof transfers the influence-function expansion to the efficient influence function for the full-information target.Source: §A.5 · p. 46

A.6. Proof of Proposition 2.

  • The restricted outcome regression and Riesz representer are L2 projections onto H, yielding restricted Riesz identities and the stated target relation.The proof uses orthogonality of each projection residual to H.Source: §A.6 · pp. 46–47

A.7. Proof of Theorem 3.

  • Theorem 3’s proof replaces the short mixed-bias identity with Proposition 2(ii); Assumption 2 controls the resulting term at oP(n^-1/2).The remaining empirical-process terms use training-sample conditioning, leading to a linear representation and asymptotic normality.Source: §A.7 · p. 47

APPENDIX B. PROOFS FOR SECTION 3

  • Appendix B collects proofs for Section 3 and uses Pn for the empirical measure on a training fold.The fold index is suppressed in this notation.Source: Appendix B · p. 48

B.1. Proof of Lemma 2.

  • For Lemma 2, the normal equation makes regression residuals orthogonal to H, and the restricted Riesz identity yields a squared-error loss difference.The proof applies the identities to functions in H.Source: §B.1 · p. 48

B.2. Derivation of (35).

  • The derivation of (35) bounds localized complexity using both the finite-dimensional span and the dictionary’s uniform envelope.For the Riesz loss, the calculation extends through mean-square continuity and the signed-hull structure.Source: §B.2 · pp. 48–49
  • The aggregation proofs apply star aggregation to finite function classes with strongly convex, sufficiently regular losses.The proof uses a pooled augmented sample of observations and folds to apply the star algorithm to fold-specific candidates.Source: §B.2 · p. 49

B.3. Proofs of Propositions 5 and 6.

This section proves Propositions 5 and 6 by establishing oracle bounds for the star estimator under squared and Riesz losses and verifying the required conditions. It develops shared localization and complexity arguments, along with tools for selection outside cross-fitting folds and chaining over localized pairs of segments.

B.3.2. Proof of Proposition 6 (Riesz loss)

  • The Riesz-loss proof uses the same shared framework and begins by imposing boundedness conditions on the candidate functions and linear functional.The loss is f(W)^2 − 2m(O,f), with m(O,·) linear; in the ATE case it represents the treated-minus-control contrast.Source: §B.3 · p. 50, §B.3.2 · p. 55

B.3. Proofs of Propositions 5 and 6

  • Strong convexity yields a localized excess-risk bound for the star estimator, and mixed-tail chaining plus peeling gives the stated log(M)/n rate.The shared argument applies to both losses, with the offset process retaining a quadratic term and segment-increment control.Source: §B.3 · pp. 51–52

B.3.1. Proof of Proposition 5 (square loss)

  • For squared loss, the proof verifies the regularity assumptions and derives an L2(P) oracle inequality for the star estimator.The result follows by substituting the squared-loss bounds into the shared inequality and using the excess-risk identity.Source: §B.3.1 · pp. 54–55

B.3. Proofs of Propositions 5 and 6.

  • The proof checks (A2), (A3), and (A4), then obtains the L2(P) oracle inequality from Lemma 2 exactly as in the squared-loss case.Source: §B.3.2 · pp. 55–56

C.1. Selection outside the folds.

  • When a tuning index is selected on the full sample, its empirical-process term requires a uniform bound over selectable indices rather than cross-fitting’s conditioning argument.Under the lemma’s conditions, the proof establishes a normal limit and consistent variance estimation.Source: §C.1 · pp. 56–57
  • The selection cost can instead be bounded by the information the selected index extracts from the evaluation fold, which is itself bounded by selection entropy.The appendix notes that this information-based bound is not used below.Source: §C.1 · p. 57

C.2. Chaining over localized pairs of segments.

  • A dyadic-grid chaining argument over localized pairs of segments proves a Bernstein-type maximal inequality.The proof bounds successive grid increments and sums them across scales.Source: §C.2 · p. 58

C.3. Proof of Proposition 7.

The proof analyzes pooled star aggregation of fold-specific candidates and establishes a high-probability oracle bound. Its fold-wise argument accounts for comparator selection that depends on training samples.

C.3. Proof of Proposition 7.

  • The pooled star estimator applies the star algorithm to stacked fold-specific candidates on augmented observation-fold data.The proof verifies convexity for square loss and addresses the Riesz loss through its Bregman divergence.Source: §C.3 · p. 59
  • A union bound over folds and comparator indices yields a high-probability oracle bound for every comparator.The proof first controls fold-wise deviations, then converts excess risk to L2(P) error.Source: §C.3 · p. 60
  • The union over comparators is needed because the best index for the fold average depends on all training samples, including that fold’s.The proof therefore cannot fix the comparator in advance of the fold.Source: §C.3 · p. 60

C.4. Proof of Theorem 4.

The proofs establish the conditions needed for Theorem 4 and extend the argument to convex-weight aggregation. They also explain how cross-fitting and orthogonality support inference for the estimable components used in sensitivity bounds.

C.4. Proof of Theorem 4.

  • The proof localizes foldwise nuisance errors at inflated radii; on that event, their product is bounded by RgRα while the theorem’s conditions are verified.The proof also checks the remaining conditions on the localized set.Source: §C.4 · pp. 60–61

C.5. Convex weights.

  • The convex-weight proof uses nets and coverings of the simplex in place of finite-menu arguments, treating regression and representer aggregation in parallel.Theorem 5 follows the proof strategy of Theorem 4 with a simplex covering replacing Lemma 8.Source: §C.5 · pp. 62–63

APPENDIX D. PROOFS FOR SECTION 4

  • For Section 4’s sensitivity bounds, cross-fitting fixes fold-specific nuisance estimates on hold-out folds, while Neyman orthogonality reduces first-stage estimation effects to second order.The appendix combines the omitted-information identity with DML inference for a small collection of estimable components.Source: Appendix D · pp. 64–65

D.1. Proof of Lemma 3.

Lemma 3 establishes orthogonal, cross-fitted expansions and joint asymptotic normality for the estimable components. Theorem 6 combines those component results to obtain asymptotic normality and consistent standard errors for interval endpoints.

D.1. Proof of Lemma 3.

  • Lemma 3’s estimable components admit orthogonal, cross-fitted expansions, and their stacked representations are jointly asymptotically normal by the multivariate CLT.The proof also develops covariance estimation for the joint components.Source: §D.1 · pp. 65, 67

D.2. Proof of Theorem 6.

  • Theorem 6’s endpoint estimators are asymptotically normal, and their estimated standard errors are consistent.The proof obtains endpoint normality by combining the component linear representations and applying the CLT.Source: §D.2 · pp. 70–71

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