Generalization Bounds for Causal Regression: Insights, Guarantees and Sensitivity Analysis

Abstract: Many algorithms have been recently proposed for causal machine learning. Yet, there is little to no theory on their quality, especially considering finite samples. In this work, we propose a theory based on generalization bounds that provides such guarantees. By introducing a novel change-of-measure inequality, we are able to tightly bound the model loss in terms of the deviation of the treatment propensities over the population, which we show can be empirically limited. Our theory is fully rigorous and holds even in the face of hidden confounding and violations of positivity. We demonstrate our bounds on semi-synthetic and real data, showcasing their remarkable tightness and practical utility.

• The paper introduces novel generalization bounds using a Pearson chi-square change-of-measure inequality to handle hidden confounders and violated assumptions.
• It leverages reweighting in outcome regression to bridge observed and complete data distributions, offering rigorous performance guarantees.
• Empirical evaluations on semi-synthetic and Parkinson’s datasets show that these bounds facilitate better model selection and reliable treatment effect estimation.

Generalization Bounds for Causal Regression: Insights, Guarantees, and Sensitivity Analysis

Introduction to Causal Machine Learning

Causal machine learning is a rapidly growing area with wide applications in fields such as economics, medicine, and education. The distinction between causal ML and traditional ML pivots on predicting not just outcomes based on covariates, but potential outcomes under different treatments. For example, consider predicting patient recovery times if a certain treatment is administered versus if it isn't. This is fundamentally different from predicting recovery times based solely on past treatments due to inherent biases in real-world data.

The core challenge in causal ML is the inability to observe both potential outcomes for the same individual. When someone receives a treatment, we can only see the outcome for the administered treatment — the counterfactual remains unknown. To tackle this, strong assumptions like ignorability and positivity are often made, often leading us into the field of sensitivity analysis when these assumptions are violated.

Understanding Generalization Bounds

One of the paper's key contributions is developing generalization bounds for causal regression. Generalization bounds offer theoretical guarantees on how well our model is expected to perform on unseen data. The novelty here is a tight change-of-measure inequality using the Pearson $\chi^2$ divergence, allowing they to bound model loss even with hidden confounding and positivity violations.

Outcome Regression

In outcome regression, they aim to predict potential outcomes given covariates. Traditional methods don't provide guarantees when assumptions like ignorability or positivity fail. This work introduces bounds that rely on reweighting samples to bridge the observed data distribution and the complete data distribution.

They use a powerful change-of-measure inequality: $$E[\mathrm{Loss}] \leq E[\mathrm{Loss} | T=a] + \lambda \cdot \Delta + \sigma^2/4\lambda$$ where $\Delta$ quantifies deviation from randomized trials, and this term can empirically be upper-bounded using observable quantities.

Individual Treatment Effect Estimation

For individual treatment effect estimation, the approach is extended to various meta-learners, which use established regressors as components. They offer bounds for T-learners, S-learners, and X-learners, deconstructing their loss into observable parts and providing finite-sample PAC-style bounds. These bounds include terms for empirical loss, divergence, and model complexity.

Practical Implications and Results

This theory isn't just academic but highly practical. They empirically evaluated their bounds on semi-synthetic and real datasets, demonstrating significant benefits:

1. Semi-Synthetic Data: They showed their bounds are substantially tighter than prior work, especially in scenarios with hidden confounders. These tighter bounds help better understand model performance when traditional assumptions don't hold.
2. Real Data on Parkinson's Disease: For the Parkinson's telemonitoring dataset, they illustrated how these bounds can affect model selection. Models appearing superior based on observable losses alone were shown to be on par with others when considering the bounds.

Future Directions

This work opens new avenues for causal ML:

• Quantile Treatment Effect Estimation: The bounds suggest that estimating conditional quantiles of treatment effects, previously deemed unfeasible, might be achievable by optimizing quantile losses in meta-learners.
• Improving Model Selection: These bounds provide a framework for more informed model selection in practical applications, accounting for unobservable biases and confounders.

Conclusion

By developing rigorous generalization bounds, this paper provides strong theoretical foundations for causal machine learning, ensuring models remain reliable even when facing hidden confounders and violated assumptions. These bounds not only validate existing algorithms but also set the stage for new methods and applications in various causal inference tasks, demonstrating their practical utility on both semi-synthetic and real-world data.