Cross-Fitted Debiasing Device

• The paper demonstrates an unbiased estimation method by using conditional cross-fitting to separate machine learning prediction from treatment effect estimation.
• It employs regression-adjusted Horvitz–Thompson estimators to remove plug-in bias from data reuse, ensuring unbiasedness even with model misspecification.
• The approach adapts to Bernoulli, complete, and stratified randomized designs, achieving asymptotic normality and efficiency with proper prediction function convergence.

Searching arXiv for the specified paper to ground the article and citation. The cross-fitted debiasing device is a conditional cross-fitting construction for machine-learning-assisted covariate adjustment in randomized experiments under a design-based inference framework. In this setting, potential outcomes and covariates are fixed, the randomization is the sole source of randomness, and the objective is unbiased estimation of the finite-population average treatment effect (ATE). The device partitions a single randomized experiment into two conditionally independent subexperiments, fits prediction functions on one fold, applies regression-adjusted Horvitz–Thompson estimation on the other fold, and then swaps roles. The resulting estimator is exactly unbiased under the design, accommodates flexible machine-learning-assisted covariate adjustment, allows for model misspecification, and is developed for Bernoulli randomized experiments, completely randomized experiments, and stratified randomized experiments (Lu et al., 21 Aug 2025).

1. Finite-population target and design-based formulation

The framework is finite-population and design-based. There are $N$ fixed units indexed by $i=1,\dots,N$, each with fixed potential outcomes $Y_i(1)$ and $Y_i(0)$, and a fixed covariate vector $x_i\in\mathbb{R}^d$. The treatment assignment is a random vector $Z=(Z_1,\dots,Z_N)\in\{0,1\}^N$, generated by a known experimental design $P_Z$. The observed outcome is

$Y_i = Z_i\,Y_i(1)+(1-Z_i)\,Y_i(0).$

The estimand is the finite-population ATE,

$$\bar\tau \;=\;\frac1N\sum_{i=1}^N\bigl\{\,Y_i(1)-Y_i(0)\bigr\}.$$

The paper works under the design-based, or randomization-only, inference framework: $\{Y_i(z),x_i\}$ are nonrandom, and the only randomness is in $i=1,\dots,N$0 (Lu et al., 21 Aug 2025). This formulation is central because the dependence structure induced by experimental designs such as complete randomization or stratified randomization generally violates the independently and identically distributed assumptions used by conventional sample-splitting and cross-fitting arguments.

Within this setup, the unadjusted Horvitz–Thompson estimator is

$i=1,\dots,N$1

By construction,

$i=1,\dots,N$2

This unbiasedness property provides the baseline from which covariate adjustment is developed. The central problem addressed by the cross-fitted debiasing device is how to preserve design-based unbiasedness while incorporating data-adaptive prediction functions.

2. Oracle adjustment, plug-in bias, and the motivation for debiasing

The starting point for adjustment is an oracle regression-adjusted Horvitz–Thompson-type estimator. If functions $i=1,\dots,N$3 were known, with $i=1,\dots,N$4, then for each arm $i=1,\dots,N$5,

$i=1,\dots,N$6

and

$i=1,\dots,N$7

The paper states that this estimator remains unbiased for any $i=1,\dots,N$8 (Lu et al., 21 Aug 2025).

In practice, however, one replaces $i=1,\dots,N$9 by estimated prediction functions $Y_i(1)$0, yielding the plug-in adjusted estimator

$Y_i(1)$1

Because $Y_i(1)$2 is trained on the same data, $Y_i(1)$3 has small finite-sample bias.

This is the immediate motivation for the debiasing device. The issue is not the form of regression adjustment itself, since the oracle-adjusted estimator is unbiased for arbitrary adjustment functions, but rather the data reuse induced by estimating $Y_i(1)$4 on the same realized experiment used for treatment effect estimation. The paper’s contribution is to remove this bias through conditional cross-fitting that is compatible with design-based randomization.

A common misconception is that ordinary cross-fitting can simply be transplanted into randomized experiments without modification. The paper explicitly contrasts its proposal with traditional sample-splitting and cross-fitting methods, which require independently and identically distributed data and therefore do not directly apply under the design-based framework when $Y_i(1)$5 are dependent across units (Lu et al., 21 Aug 2025).

3. Conditional cross-fitting algorithm

The core construction is Algorithm 1, termed conditional cross-fitting. Its stated goal is to partition the experiment into two conditionally independent subexperiments, fit on one, and estimate on the other.

The algorithm begins with a random split $Y_i(1)$6 of $Y_i(1)$7, generated through a mechanism $Y_i(1)$8 satisfying Assumption A.1. That assumption has two components:

1. $Y_i(1)$9 is independent of $Y_i(0)$0 given $Y_i(0)$1.
2. $Y_i(0)$2 for all $Y_i(0)$3.

For each fold $Y_i(0)$4, the procedure fits prediction functions $Y_i(0)$5 using only the opposite fold $Y_i(0)$6. In general, the fitted function solves

$Y_i(0)$7

where $Y_i(0)$8 may be any loss, including squared error, logistic, tree loss, and related losses.

The fitted functions are then used on the hold-out fold $Y_i(0)$9 to construct the covariate-adjusted estimator

$x_i\in\mathbb{R}^d$0

with

$x_i\in\mathbb{R}^d$1

The final estimator is the weighted combination

$x_i\in\mathbb{R}^d$2

The paper states that, conditional on $x_i\in\mathbb{R}^d$3, the two summands are independent unbiased estimators of $x_i\in\mathbb{R}^d$4, yielding Theorem 4.1: $x_i\in\mathbb{R}^d$5 This is the debiasing mechanism proper: training and estimation are separated across folds, but the separation is engineered so that it respects the randomization design rather than an i.i.d. sampling model (Lu et al., 21 Aug 2025).

4. Randomization-specific splitting schemes

The validity of conditional cross-fitting depends on how the split is constructed. The paper gives explicit sample-splitting algorithms for several standard randomized designs.

Design	Splitting scheme	Conditional property
Bernoulli randomized experiment	Draw independent $x_i\in\mathbb{R}^d$6; let $x_i\in\mathbb{R}^d$7, $x_i\in\mathbb{R}^d$8	Conditional on $x_i\in\mathbb{R}^d$9, the two subsamples are independent Bernoulli trials of sizes $Z=(Z_1,\dots,Z_N)\in\{0,1\}^N$0 with the same $Z=(Z_1,\dots,Z_N)\in\{0,1\}^N$1
Completely randomized experiment	Split treated and control sets separately into $Z=(Z_1,\dots,Z_N)\in\{0,1\}^N$2 and $Z=(Z_1,\dots,Z_N)\in\{0,1\}^N$3; define $Z=(Z_1,\dots,Z_N)\in\{0,1\}^N$4	Conditioned on $Z=(Z_1,\dots,Z_N)\in\{0,1\}^N$5, each fold is itself a CRE of size $Z=(Z_1,\dots,Z_N)\in\{0,1\}^N$6 with $Z=(Z_1,\dots,Z_N)\in\{0,1\}^N$7 treated
Stratified randomized experiment	Either split-by-treatment within each stratum, or split strata into two whole sets $Z=(Z_1,\dots,Z_N)\in\{0,1\}^N$8	Each fold is conditionally an independent SRE

For a Bernoulli randomized experiment with $Z=(Z_1,\dots,Z_N)\in\{0,1\}^N$9, Example 3.1 uses an independent Bernoulli split $P_Z$0. Proposition 3.2 then states that, conditional on $P_Z$1, the two subsamples are independent Bernoulli trials of sizes $P_Z$2 with the same $P_Z$3.

For a completely randomized experiment with $P_Z$4, Example 3.3 adopts split-by-treatment. Positive integers $P_Z$5 are chosen to sum to $P_Z$6; within each arm $P_Z$7, the set $P_Z$8 is randomly split into $P_Z$9 and $Y_i = Z_i\,Y_i(1)+(1-Z_i)\,Y_i(0).$0; and the fold is $Y_i = Z_i\,Y_i(1)+(1-Z_i)\,Y_i(0).$1. Proposition 3.4 states that, conditioned on $Y_i = Z_i\,Y_i(1)+(1-Z_i)\,Y_i(0).$2, each fold is itself a CRE of size $Y_i = Z_i\,Y_i(1)+(1-Z_i)\,Y_i(0).$3 with $Y_i = Z_i\,Y_i(1)+(1-Z_i)\,Y_i(0).$4 treated.

For a stratified randomized experiment with $Y_i = Z_i\,Y_i(1)+(1-Z_i)\,Y_i(0).$5 strata, the paper gives two options: split-by-treatment within each stratum, which requires at least 2 units per arm in each stratum, or split the strata into two whole sets $Y_i = Z_i\,Y_i(1)+(1-Z_i)\,Y_i(0).$6. Propositions 3.5–3.6 show that each fold is conditionally an independent SRE (Lu et al., 21 Aug 2025).

These schemes are not ancillary implementation choices. They are the mechanism by which conditional independence of the fold-specific assignments is guaranteed, and therefore they are the design-based substitute for the i.i.d. structure assumed in conventional cross-fitting.

5. Unbiasedness, asymptotic normality, and inference

The paper’s theoretical guarantees are organized around unbiasedness, asymptotic normality, efficiency, and valid confidence intervals.

Unbiasedness is exact and finite-sample. Under Assumption A.1,

$Y_i = Z_i\,Y_i(1)+(1-Z_i)\,Y_i(0).$7

and therefore

$Y_i = Z_i\,Y_i(1)+(1-Z_i)\,Y_i(0).$8

This exact design-based unbiasedness is the defining property of the debiasing device (Lu et al., 21 Aug 2025).

For asymptotic normality and efficiency, the paper states that under mild stability conditions,

$Y_i = Z_i\,Y_i(1)+(1-Z_i)\,Y_i(0).$9

fold variances bounded as in Assumption A.2, and a nondegenerate limit,

$$\bar\tau \;=\;\frac1N\sum_{i=1}^N\bigl\{\,Y_i(1)-Y_i(0)\bigr\}.$$0

The limiting variance $$\bar\tau \;=\;\frac1N\sum_{i=1}^N\bigl\{\,Y_i(1)-Y_i(0)\bigr\}.$$1 equals that of the oracle regression-adjusted estimator $$\bar\tau \;=\;\frac1N\sum_{i=1}^N\bigl\{\,Y_i(1)-Y_i(0)\bigr\}.$$2.

Variance estimation is built fold-wise from cross-fitted residuals

$$\bar\tau \;=\;\frac1N\sum_{i=1}^N\bigl\{\,Y_i(1)-Y_i(0)\bigr\}.$$3

The paper proposes conservative fold-wise variance estimators $$\bar\tau \;=\;\frac1N\sum_{i=1}^N\bigl\{\,Y_i(1)-Y_i(0)\bigr\}.$$4 by applying standard design-based estimators to these residuals, and then combines them as

$$\bar\tau \;=\;\frac1N\sum_{i=1}^N\bigl\{\,Y_i(1)-Y_i(0)\bigr\}.$$5

Theorem 5.3 states that if each $$\bar\tau \;=\;\frac1N\sum_{i=1}^N\bigl\{\,Y_i(1)-Y_i(0)\bigr\}.$$6 is consistent or conservative for the oracle variance and $$\bar\tau \;=\;\frac1N\sum_{i=1}^N\bigl\{\,Y_i(1)-Y_i(0)\bigr\}.$$7 is asymptotically Normal, then

$$\bar\tau \;=\;\frac1N\sum_{i=1}^N\bigl\{\,Y_i(1)-Y_i(0)\bigr\}.$$8

has asymptotic coverage $$\bar\tau \;=\;\frac1N\sum_{i=1}^N\bigl\{\,Y_i(1)-Y_i(0)\bigr\}.$$9.

The paper also states an efficiency result for splitting. Among all splits satisfying a mild balance condition, the one with

$\{Y_i(z),x_i\}$0

attains the same asymptotic variance as the oracle-adjusted estimator and thus maximizes efficiency. This is given as Theorem 4.2 (Lu et al., 21 Aug 2025).

A plausible implication is that the random split is part of the asymptotic design problem rather than a purely computational convenience. In the formulation of the paper, balanced conditional treatment probabilities across folds are directly tied to first-order efficiency.

6. Flexible machine learning, misspecification, and relation to i.i.d. cross-fitting

The role of machine learning in the construction is deliberately abstract. The prediction rule $\{Y_i(z),x_i\}$1 may be any estimator in a rich function class $\{Y_i(z),x_i\}$2, including trees, lasso, neural nets, and boosting. The loss $\{Y_i(z),x_i\}$3 is similarly unrestricted in the abstract algorithm, and may be squared error, logistic, tree loss, or other losses (Lu et al., 21 Aug 2025).

The paper makes three separate claims about robustness to the prediction stage. First, unbiasedness of $\{Y_i(z),x_i\}$4 holds irrespective of the quality of $\{Y_i(z),x_i\}$5. Second, asymptotic variance reduction requires only that $\{Y_i(z),x_i\}$6 converges in $\{Y_i(z),x_i\}$7 to some limit $\{Y_i(z),x_i\}$8, described as the stability Assumption A.3. Third, no correct model form is required: even if $\{Y_i(z),x_i\}$9 is misspecified, $i=1,\dots,N$00 remains unbiased, and efficiency gains accrue to the extent that $i=1,\dots,N$01 captures variation in $i=1,\dots,N$02.

This distinguishes the method from regression adjustment procedures whose validity is tied to correct specification. Here, the design-based unbiasedness argument is detached from predictive correctness. The machine-learning stage affects precision and asymptotic efficiency, not the design-based centering of the estimator.

The contrast with i.i.d. cross-fitting is explicit. Under design-based inference, $i=1,\dots,N$03 are fixed constants and only $i=1,\dots,N$04 is random. There is no i.i.d. assumption on $i=1,\dots,N$05 or on $i=1,\dots,N$06. Indeed, under completely randomized or stratified randomized experiments, $i=1,\dots,N$07 are dependent across units. The only critical assumption for unbiasedness is conditional independence of $i=1,\dots,N$08 given the split, and the paper’s specialized splitting schemes are designed to guarantee it.

A common misunderstanding is to view the procedure as merely a standard sample split adapted to experimentation. The paper’s formulation is narrower and more specific: it is a conditional independence construction under randomization-only inference. This suggests that the “debiasing device” is best understood not as generic cross-validation-style sample splitting, but as a design-compatible transformation of a randomized experiment into two subexperiments whose independence is conditional on the realized split.