The theoretical guarantees on the consistency and convergence rates of the final weighted estimator derived from the proposed representation learning approach are grounded in the framework's ability to minimize the balancing score error (BSE) and the bias associated with the chosen representation. Under the assumptions outlined in the paper, including the absolute continuity of the target distribution with respect to the source distribution, the estimator is shown to be consistent as the sample size increases.
Specifically, the convergence rates can be established through the analysis of the bias decomposition, which includes terms for the bias with respect to the representation and the confounding bias. The proposed method provides a more flexible framework compared to existing methods, as it does not rely on strict assumptions about the outcome model or the representation being well-specified. This flexibility allows for better performance in practical scenarios where such assumptions may not hold.
In comparison to existing methods, such as those relying on propensity score matching or outcome regression, the proposed approach offers improved robustness against model misspecification. While traditional methods often require well-defined models for both the outcome and the treatment assignment, the representation learning framework can adaptively learn representations that minimize bias without direct access to outcome information, leading to potentially lower bias and variance in the final estimates.