Orthogonal Uplift Learning with Permutation-Invariant Representations for Combinatorial Treatments

TL;DR

An uplift estimation framework that aligns treatment representation with causal semantics is proposed, extending Robinson-style decompositions to learned, vector-valued treatments and is shown to be expressive for policy-induced causal effects, orthogonally robust to nuisance estimation errors, and stable under small policy perturbations.

Abstract

We study uplift estimation for combinatorial treatments. Uplift measures the pure incremental causal effect of an intervention (e.g., sending a coupon or a marketing message) on user behavior, modeled as a conditional individual treatment effect. Many real-world interventions are combinatorial: a treatment is a policy that specifies context-dependent action distributions rather than a single atomic label. Although recent work considers structured treatments, most methods rely on categorical or opaque encodings, limiting robustness and generalization to rare or newly deployed policies. We propose an uplift estimation framework that aligns treatment representation with causal semantics. Each policy is represented by the mixture it induces over contextaction components and embedded via a permutation-invariant aggregation. This representation is integrated into an orthogonalized low-rank uplift model, extending Robinson-style decompositions to learned, vector-valued treatments. We show that the resulting estimator is expressive for policy-induced causal effects, orthogonally robust to nuisance estimation errors, and stable under small policy perturbations. Experiments on large-scale randomized platform data demonstrate improved uplift accuracy and stability in long-tailed policy regimes

Figures (1)

• Figure 1: Treatment Net: permutation-invariant embedding of combinatorial treatments. A treatment $T$ is observed through its policy specification: for each context $s\in S$ (instantiated here as $\{s_1, s_2\}$), the treatment induces a distribution over atomic actions $a\in \mathcal{A}$ (instantiated as $\{a_0, a_1, a_2\}$). The network first learns embeddings for contexts and actions to form atom representations $\phi(s,a)$. These atoms are then reweighted by policy probabilities and aggregated via a permutation-invariant sum $z(T)$, ensuring the representation depends only on the induced mixture. Finally, a learnable map produces the treatment embedding $h(T)$, which is used by the orthogonalized uplift model (left panel) to estimate the causal effect $\tau(x)$.

Theorems & Definitions (12)

• Proposition 3.2: Well-specified orthogonalized low-rank representation
• Lemma 3.3: Expressiveness of permutation-invariant embeddings
• Lemma 3.4: Neyman orthogonality w.r.t. nuisance $(m,e)$
• Theorem 3.6: Orthogonal robustness: nuisance errors enter at second order
• Proposition 3.7: Stability of the embedding and uplift
• proof : Proof of Proposition \ref{['prop:olr_wellspec']}
• proof : Proof of Lemma \ref{['thm:expressiveness']}
• proof : Proof of Lemma \ref{['lem:neyman_orth']}
• Lemma 2.1: Second-order control of score perturbation
• proof : Proof of Lemma \ref{['lem:score_perturb']}