Leakage in X-Learner in-sample prediction

[kklein]

Issue at hand

@ArseniyZvyagintsevQC brought the following to our attention:

Let us assume a binary treatment variant scenario in which we want to work with in-sample predictions, i.e. is_oos=False.

The current implementation would go about fitting five models, three of which considered nuisance models and two of which considered treatment models:

model	target	cross-fitting dataset	stage	name
$\hat{\mu}_0$	$Y_i$	$\{(X_i, Y_i) | W_i=0\}$	nuisance	"treatment_variant"
$\hat{\mu}_1$	$Y_i$	$\{(X_i, Y_i) | W_i=1\}$	nuisance	"treatment_variant"
$\hat{e}$	$W_i$	$\{(X_i, Y_i)\}$	nuisance/propensity	"propensity_model"
$\hat{\tau}_0$	$\hat{\mu}(X_i) - Y_0$	$\{(X_i, Y_i) | W_i=0\}$	treatment	"control_effect_model"
$\hat{\tau}_1$	$Y_i - \hat{\mu}(X_i)$	$\{(X_i, Y_i) | W_i=1\}$	treatment	"treatment_effect_model"

More background on this here.

Note that each of these models is cross-fitted. More precisely, each is cross-fitted wrt the data it has seen at training time.

Let's suppose now that we are at inference time and encounter an in-sample data point $i$. Wlog, let's assume that $W_i=1$.
In order to come up with a CATE estimate, the predict method will run

• $\hat{\tau}_0(X_i)$ with is_oos=True since this datapoint has not been seen during training time of the model $\hat{\tau}_0$
• $\hat{\tau}_1(X_i)$ with is_oos=False since this datapoint has indeed been seen during the training time of the model $\hat{\tau}_1$

The latter call makes sure we avoid leakage in $\hat{\tau}_1$. The former call, however, does not completely avoid leakage:
even though $i$ hasn't been seen in the training of $\hat{\tau}_0$, it has been seen in $\hat{\mu}_1$, which is, in turn, used by $\hat{\tau}_0$. Therefore, the observed outcome $Y_i$ can leak into the estimate $\hat{\tau}(X_i)$.

Next steps

We can devise an extreme, naïve approach to counteract this issue by training every type of model once per datapoint. Clearly, this ensures the absence of data leakage. The challenge with this issue revolves around coming up with a design that

• allows for arbitrary numbers (>1, <=n) of cross-fitting folds, i.e. not fixing it to be equal to the number of training data points
• integrates well into the structure of the library

[kklein]

Preliminary idea

Currently we are training

• n_folds many $\hat{\mu}_0$ models
• n_folds many $\hat{\mu}_k$ models for every $k$
• n_folds many $\hat{\tau}_{0,k}$ models for every $k$
• n_folds many $\hat{\tau}_{k,0}$ models for every $k$

In order to answer an in-sample query of

Give me models $\hat{\tau}_{0,k}$ and $\hat{\tau}_{k,0}$ which have seen no information about sample $i$ at all

We could train

• n_folds * n_folds many $\hat{\tau}_{0,k}$ models for every $k$
• n_folds * n_folds many $\hat{\tau}_{k,0}$ models for every $k$

In the scenario described in the issue, we would then run the predict method as such:

• $\hat{\tau}_0(X_i)$ is estimated by fetching the n_folds * (n_folds - 1) many models $\hat{\tau}_0(X_i)$ which are based on $\hat{\mu}_k$ models, which have not seen data point $i$; these model estimates can be aggregated
• $\hat{\tau}_1(X_i)$ is estimated by fetching the n_folds * (n_folds - 1) many model $\hat{\tau}_1(X_i)$ which have not seen $i$ and have used any $\hat{\mu}_0$ models; these model estimates can be aggregated

Importantly, this would

• Redefine the meaning of is_oos
• Massively increase the computational burden