GDR-learners: Orthogonal Learning of Generative Models for Potential Outcomes

TL;DR

This paper introduces a general suite of generative Neyman-orthogonal (doubly-robust) learners that estimate the conditional distributions of potential outcomes and possesses the properties of quasi-oracle efficiency and rate double robustness, and are thus asymptotically optimal.

Abstract

Various deep generative models have been proposed to estimate potential outcomes distributions from observational data. However, none of them have the favorable theoretical property of general Neyman-orthogonality and, associated with it, quasi-oracle efficiency and double robustness. In this paper, we introduce a general suite of generative Neyman-orthogonal (doubly-robust) learners that estimate the conditional distributions of potential outcomes. Our proposed generative doubly-robust learners (GDR-learners) are flexible and can be instantiated with many state-of-the-art deep generative models. In particular, we develop GDR-learners based on (a) conditional normalizing flows (which we call GDR-CNFs), (b) conditional generative adversarial networks (GDR-CGANs), (c) conditional variational autoencoders (GDR-CVAEs), and (d) conditional diffusion models (GDR-CDMs). Unlike the existing methods, our GDR-learners possess the properties of quasi-oracle efficiency and rate double robustness, and are thus asymptotically optimal. In a series of (semi-)synthetic experiments, we demonstrate that our GDR-learners are very effective and outperform the existing methods in estimating the conditional distributions of potential outcomes.

Figures (8)

• Figure 1: Capturing the uncertainty in POs helps make reliable treatment decisions. Unlike conditional average potential outcomes (CAPOs, left), conditional distributions of potential outcomes (CDPOs, right) allow for quantifying the aleatoric uncertainty of the POs. As a result, they capture more information about the potential outcomes, such as heavy tails or multi-modalities.
• Figure 2: Difference to the IPTW-learner. Comparison of the IPTW-learners (= a variant of our GDR-learners with the target model class $\xi_a^* \in \mathit{\Xi}$) and our original GDR-learners (with the target model class $g_a^* \in \mathcal{G}$) when $V = X$ and $\mathcal{G} \subset \mathit{\Xi}$. Here, we show three scenarios depending on whether the ground-truth CDPOs, $\textcolor{red}{\bigstar} = \mathbb{P}(Y[a] \mid x)$, belong to both $\mathcal{G}$ and $\mathit{\Xi}$, only $\mathit{\Xi}$, or neither of both. In Scenarios 1 and 2, both the IPTW-learner and our GDR-learners are Neyman-orthogonal and, therefore, quasi-oracle efficient. In Scenarios 3, on the other hand, none of the learners are Neyman-orthogonal, as $\textcolor{red}{\bigstar} = \xi_a \notin \mathit{\Xi}$. However, in Scenario 1, our GDR-learners have a smaller empirical error as $\mathcal{G} \subset \mathit{\Xi}$. Thus, we should prefer the IPTW-learners in Scenario 2 and our GDR-learners in Scenario 1. Note that the real scenario is unknown in practice.
• Figure 3: Overview of our GDR-learners. Our GDR-learners proceed in two stages. In the first stage, nuisance conditional generative models are trained to estimate the nuisance functions $\hat{\eta} = (\hat{\xi}_a, \hat{\pi}_a)$ jointly for $a \in \{0, 1\}$. In the second stage, target conditional generative models use the outputs of the nuisance conditional generative models to optimize the $\hat{\mathcal{L}}_{\text{GDR}}$ wrt. $g_0$ and $g_1$, see Eq. \ref{['eq:gdr-learner']}.
• Figure 4: Results for the synthetic experiments with varying size of training data ($n_{\text{train}}$). Reported: mean out-sample $W_2 \, \pm$ se over 20 runs (lower is better).
• Figure 5: Results for our GDR-learners on the synthetic experiments with varying size of training data ($n_{\text{train}}$) and varying strength of EMA smoothing ($\lambda$). Reported: mean out-sample $W_2 \, \pm$ se over 20 runs (lower is better).

Theorems & Definitions (17)

• Theorem 1: Neyman-orthogonality
• proof
• Theorem 2: Quasi-oracle efficiency and double robustness
• proof
• Definition 1: Neyman-orthogonality foster2023orthogonalmorzywolek2023general
• Definition 2: Quasi-oracle efficiency
• Definition 3: Star hull
• Lemma 1: Identification of the target risks
• proof
• Lemma 2: One-step bias correction of the RA-learner