The DR-Learner

The T-learner's estimate of a treatment effect is only as good as its two outcome models — if either is badly misspecified (missing an important nonlinearity, say), the estimated effect can be badly biased with no warning sign. A separate approach to estimating treatment assignment probability, the propensity score, has the same vulnerability if it's wrong. What if you could combine both kinds of model so that the combination is protected against either one alone being wrong? That's the promise behind "doubly robust" estimation, and the DR-learner is the meta-learner that builds it into individual-level treatment effect estimation.

The idea: two models, combined so either can fail alone

The DR-learner uses two auxiliary models: an outcome model $\hat{\mu}_t(x)$ (predicting the outcome under treatment $t$, as in a T-learner) and a propensity model $\hat{e}(x)$ (predicting the probability of receiving treatment given $x$). It combines them into a "pseudo-outcome" for each unit,

$$\hat{\psi}_i = \hat{\mu}_1(x_i) - \hat{\mu}_0(x_i) + \frac{T_i (Y_i - \hat{\mu}_1(x_i))}{\hat{e}(x_i)} - \frac{(1-T_i)(Y_i - \hat{\mu}_0(x_i))}{1 - \hat{e}(x_i)}$$

and then regresses this pseudo-outcome on $x$ to get the final treatment-effect estimate $\hat{\tau}(x)$. In plain English: the first two terms are the plain T-learner-style difference in predicted outcomes; the remaining two terms are a correction that reweights each unit's actual residual — how wrong the outcome model was for that specific unit — by the inverse of its propensity score. If the outcome models $\hat\mu_1, \hat\mu_0$ are exactly right, the residual terms average to zero and the formula reduces to the T-learner. If instead the propensity model $\hat e$ is exactly right, the correction terms exactly cancel out any bias in a wrong outcome model. The estimate is consistent as long as at least one of the two auxiliary models is correctly specified — the "doubly robust" property.

Worked example: one wrong model, still a good estimate

Suppose the true treatment effect for a certain order type is exactly 10 (in fill-quality points), and the outcome model is slightly biased, systematically predicting fill quality 3 points too low for both treated and control units (say, because it's missing a relevant feature). A plain T-learner using only these biased outcome models would estimate the effect as $\hat\mu_1(x) - \hat\mu_0(x)$ — since both predictions are off by the same constant amount, the bias actually cancels in this particular case, but in general a biased outcome model doesn't cancel so neatly, and could easily distort the difference. The DR-learner's residual-correction terms explicitly measure how wrong $\hat\mu_1$ and $\hat\mu_0$ were on the actual observed data (via $Y_i - \hat\mu_1(x_i)$ and $Y_i - \hat\mu_0(x_i)$) and feed that error back in, weighted by the propensity score — so even when the outcome model's bias doesn't happen to cancel, as long as the propensity model $\hat e(x)$ is estimated well, the correction terms pull the final pseudo-outcome back toward the true value of 10.

What this means in practice

The DR-learner is the standard choice when a practitioner is unsure whether their outcome model or their propensity model is better specified — a common situation in trading and execution settings, where the "true" relationship between order features and fill quality, or between order features and which algorithm gets chosen, is rarely known with confidence. Its robustness comes at the cost of complexity: it requires fitting three models in total (two outcome models, one propensity model) and, critically, these auxiliary models should be fit on data separate from the pseudo-outcome regression itself — usually via cross-fitting — or the doubly robust guarantee breaks down.