Doubly Robust Estimation of Causal Effect on CVR with Targeted Regularization

{ "core_finding": "The authors develop a doubly robust causal effect estimator and a targeted regularization framework specifically for chain-structured outcomes like post-click conversion rates, achieving faster convergence rates than nuisance parameter estimation and robust performance against model misspecification.", "caveats": "The proposed estimator relies on positivity and unconfoundedness assumptions, and its theoretical guarantees depend on nuisance estimators converging at rates faster than n^{-1/4}.", "markdown": "## Research Motivation and Challenges\n\nEstimating the causal effect of interventions on the post-click conversion rate (CVR) is critical in digital advertising and e-commerce, as strategies like high-value coupon allocation can simultaneously increase click probability and alter post-click conversion efficiency. Restricting causal analysis solely to clicked samples introduces severe sample selection bias because click behavior is non-random. While prior CVR prediction studies introduce ideal loss functions to debias the loss over full samples, there is no formal guarantee that an unbiased loss yields an unbiased final causal estimator. Furthermore, applying standard causal estimators directly fails because the second-stage conversion outcome is strictly dependent on the first-stage click outcome.\n\n## Semiparametric Theory and Doubly Robust Estimation\n\nTo overcome these limitations, the authors revisit CVR causal estimation from a semiparametric perspective. By deriving the influence function and the von Mises expansion for the target estimand, they construct a novel doubly robust estimator. This estimator remains consistent even if some nuisance estimators are misspecified and achieves a root-n convergence rate, which is faster than the convergence rates of the individual nuisance parameters. This decouples the final estimation quality from the slow convergence typical of flexible nonparametric models like neural networks.\n\n## Targeted Regularization Framework\n\nAlthough the theoretical one-step doubly robust correction works asymptotically, it can suffer from finite-sample instability due to small click-through rates in the denominator and the complexities of cross-fitting. To resolve this, the authors introduce a practical estimation framework driven by targeted regularization. By incorporating a specialized regularizer that forces the influence function correction term close to zero while learning a perturbation parameter, the framework enhances numerical stability and practical applicability for continuous and discrete treatments.\n\n## Key Terms and Definitions\n- Post-click conversion rate (CVR) — The conditional probability that a conversion occurs given that a click has occurred under a specific treatment.

• Ideal loss — A theoretical loss function designed to optimize model parameters using an unbiased estimate evaluated over the entire population sample.
• Nuisance parameters — Intermediary functions such as propensity scores and conditional expectations that must be estimated but are not the primary causal targets of interest.
• Influence function — A directional derivative over the distribution space that characterizes the asymptotic behavior and bias properties of a regular estimator.
• Targeted regularization — A training technique that penalizes violations of the efficient influence curve equation to improve finite-sample numerical stability." }