The S-Learner and T-Learner

Suppose you want to know how much a trading desk's move to a new execution algorithm improved fill quality for each individual order, not just on average — because some order types might benefit a lot and others not at all, and you'd like to route accordingly. You have historical data on orders that did and didn't use the new algorithm, plus features describing each order (size, venue, time of day). The meta-learner framework turns this into an off-the-shelf supervised learning problem, and the S-learner and T-learner are the two simplest ways to set it up.

The idea: predict outcomes under each treatment, then difference them

Both approaches want the same quantity, the individual-level treatment effect given features $x$:

$$\tau(x) = \mathbb{E}[Y \mid X = x, T = 1] - \mathbb{E}[Y \mid X = x, T = 0]$$

In plain English: $\tau(x)$ is the expected outcome difference between being treated ($T=1$) and not ($T=0$) for a unit with features $x$ — the effect of the new execution algorithm, say, on an order with exactly this size and venue. The two learners differ only in how they estimate the two conditional expectations on the right-hand side.

The S-learner (single model) trains one model $\hat{\mu}(x, t)$ on all the data, with the treatment indicator $t$ folded in as just another input feature alongside $x$. To estimate the effect for a given $x$, it predicts once with $t=1$ and once with $t=0$ and takes the difference: $\hat{\tau}(x) = \hat{\mu}(x, 1) - \hat{\mu}(x, 0)$.

The T-learner (two models) instead splits the data by treatment group and trains two entirely separate models — $\hat{\mu}_1(x)$ on treated units only, $\hat{\mu}_0(x)$ on untreated units only — then differences their predictions: $\hat{\tau}(x) = \hat{\mu}_1(x) - \hat{\mu}_0(x)$.

Worked example: a small execution-quality dataset

Ten large orders used the new algorithm (treated), ten used the old one (control), each with a "size" feature. Fitting a T-learner means training a model on the ten treated orders alone to predict fill quality from size, and a completely separate model on the ten control orders alone. If the treated-only model predicts a fill quality of 92 for a size-500 order and the control-only model predicts 85 for the same size, the T-learner's estimated effect is $\hat{\tau}(500) = 92 - 85 = 7$. An S-learner instead fits one model on all twenty orders with size and a treatment flag as joint inputs; if that shared model is a tree-based method with modest depth, and treatment barely improves its splits relative to size, it may end up effectively ignoring the treatment flag and predicting almost the same fill quality regardless of $t$ — understating the effect because the single model, sharing structure across both groups, can end up "regularizing away" a genuine treatment effect it never gets to model with enough dedicated flexibility.

What this means in practice

The T-learner is preferred when treatment and control groups plausibly have quite different response surfaces and there's enough data in each group to fit separately; it can suffer, though, when one group is much smaller than the other, since the smaller group's model is trained on less data and becomes noisier. The S-learner is preferred when data is scarce and a shared model can borrow strength across both groups, but it risks the opposite failure — a regularized or low-flexibility model can implicitly treat the treatment indicator as unimportant relative to other features, shrinking real effects toward zero. Neither correctly accounts for confounding on its own; both assume treatment assignment is as good as random once conditioning on the observed features, which is a strong assumption in most financial and trading settings and should be checked before trusting the estimated effects.