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I am reading the double-machine learning paper by Chernozhukov et al. (2018), in Example 1.1. the authors consider the partially linear model:

$$Y = D \theta_0 + g_0(X) + U, E[U|X, D] = 0\\ D = m_0(X) + V, E[V|X] = 0$$

and then they're saying:

"A naive approach to estimation of $\theta_0$ using ML methods would be, for example, to construct a sophisticated ML estimator $D\hat{\theta}_0 + \hat{g}_0(X)$ for learning the regression function $D \theta_0 + g(X)$."

Then they simulate this approach and provide the following picture of the empirical distribution of $\hat{\theta}_0 - \theta_0$ based on simulations,enter image description here illustrating that $\hat{\theta}_0$ is biased.

Does anybody have an idea of what kind of "sophisticated ML estimator" they used to get such results?

More generally, how would you estimate $\theta_0$ and $g(X)$ using standard ML methods?

What first came to my mind was to start with some guess for $\theta_0$, call it $\tilde{\theta}$ then run LightGBM regressor of $Y - D\tilde{\theta}$ on $X$, then compute a prediction $\hat{g}_1(X)$, compute $\hat{Y}:=Y - \hat{g}_1(X)$ and run a simple OLS of $\hat{Y}$ on $D$ getting a new value for $\tilde{\theta}$, then iterate until convergence.

For simulations, I took the same parameters $n=500, p=20$, $X \sim U[0, 1]^p$. For functions I took $g_0(X) = 0.2+\sin(2 \pi X_1) + \frac{1}{1+e^{10*X_2}} + 0.4\max\{X_3, 0.3\}$, $D \sim Bern(\min\{\max\{0.3*X_1-X_2^2+X_4/3, 0.3\}, 1\})$ (just random non-linear/non-smooth functions that came to my mind). Then I ran the above algorithm with LGB with default parameters, and got the following results (this is also the distribution of $\hat{\theta}_0 - \theta$ based on 700 Monte Carlo simulations. enter image description here This doesn't at all look as biased as in the paper. I am sure what I was doing must be not the smartest thing to do, and there should be better standard methods, but I am a bit confused by the fact that even this simple method that didn't require any involved cross-fitting/orthogonalization/etc., as well as any tuning, worked not as bad as the paper says it should.

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I think I figured the answer to my own question. Apparently, what they did was estimating $E[Y|D, X]$ (i.e. fitting a model using $(D, X)$ as features), and then computing the estimator of $\theta_0$ as $$\hat{\mathbb{E}}[Y|X, D=1] - \hat{\mathbb{E}}[Y|X, D=0]$$ and average the above quantity over $X$. Simulations (below) gave very similar (in terms of biasedness) distribution to that in the paper.

While a much better method of estimating $\theta_0$ was proposed by Kunzel et al. (2019) that they called "X-learner". The idea is that you use a subsample $D=1$ to estimate $\mu_1(X):=\mathbb{E}[Y|X, D=1]$, a subsample $D=0$ to estimate $\mu_0(X):=\mathbb{E}[Y|X, D=0]$, then for units with $D=1$ you compute a predicted $\hat{\mu}_0(X)$, and symetrically for units with $D=0$ you compute a predicted $\hat{\mu}_1(X)$. Finally, $$\hat{\theta}_0 = \frac{1}{n}\sum_{i:D_i = 1} (Y_i - \hat{\mu}_0(X_i)) + \frac{1}{n}\sum_{i:D_i = 0} (\hat{\mu}_1(X_i) - Y_i)$$

Below are the results of simulations: enter image description here

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