Beyond Consistency: Inference for the Relative risk functional in Deep Nonparametric Cox Models

2026-03-24

AI Summary

This work addresses the lack of reliable statistical inference theory for deep neural networks in nonparametric Cox models, particularly regarding optimization error propagation, bias control, and uncertainty quantification. It establishes, for the first time, pointwise and multivariate asymptotic normality for deep Cox estimators by linking optimization error to excess risk through non-asymptotic oracle inequalities. The approach leverages structured neural network parameterizations to control approximation bias and integrates subsampling ensembles, Hajek–Hoeffding projections, and the infinitesimal jackknife to enable valid Wald-type inference for relative risk functionals such as log hazard ratios. The theoretical analysis relaxes conventional covariance decay conditions, and both simulations and real-data experiments demonstrate the method’s practical utility in finite samples.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• User Modeling, Personalization and Recommendation: Practical large-scale studies of user experience
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

There remain theoretical gaps in deep neural network estimators for the nonparametric Cox proportional hazards model. In particular, it is unclear how gradient-based optimization error propagates to population risk under partial likelihood, how pointwise bias can be controlled to permit valid inference, and how ensemble-based uncertainty quantification behaves under realistic variance decay regimes. We develop an asymptotic distribution theory for deep Cox estimators that addresses these issues. First, we establish nonasymptotic oracle inequalities for general trained networks that link in-sample optimization error to population risk without requiring the exact empirical risk optimizer. We then construct a structured neural parameterization that achieves infinity-norm approximation rates compatible with the oracle bound, yielding control of the pointwise bias. Under these conditions and using the Hajek--Hoeffding projection, we prove pointwise and multivariate asymptotic normality for subsampled ensemble estimators. We derive a range of subsample sizes that balances bias correction with the requirement that the Hajek--Hoeffding projection remain dominant. This range accommodates decay conditions on the single-overlap covariance, which measures how strongly a single shared observation influences the estimator, and is weaker than those imposed in the subsampling literature. An infinitesimal jackknife representation provides analytic covariance estimation and valid Wald-type inference for relative risk contrasts such as log-hazard ratios. Finally, we illustrate the finite-sample implications of the theory through simulations and a real data application.

Problem

• deep Cox model
• relative risk
• asymptotic inference
• bias control
• uncertainty quantification

Innovation

• Deep Cox model
• Asymptotic normality
• Subsampling ensemble
• Infinitesimal jackknife
• Oracle inequality