Self-Consistent Equation-guided Neural Networks for Censored Time-to-Event Data Sehwan Kim 1, Rui Wang 1,2, and Wenbin Lu 3 1 Department of Population Medicine, Harvard Pilgrim Health Care Institute and Harvard Medical School, Boston, MA 2 Department of Biostatistics, Harvard School of Public Health, Boston, MA 3 Department of Statistics, North Carolina State University, Raleigh, NC March 13, 2025 Abstract In survival analysis, estimating the conditional survival function given predictors is often of interest. There is a growing trend in the development of deep learning methods for analyzing censored time-to-event data, especially when dealing with high-dimensional predictors that are complexly interrelated. Many existing deep learning approaches for estimating the conditional survival functions extend the Cox regression models by replacing the linear function of predictor effects by a shallow feed-forward neural network while maintaining the proportional hazards assumption. Their implementation can be computationally intensive due to the use of the full dataset at each iteration because the use of batch data may distort the at-risk set of the partial likelihood function. To overcome these limitations, we propose a novel deep learning approach to non-parametric estimation of the conditional survival functions using the generative adversarial networks leveraging self-consistent equations. The proposed method is model-free and does not require any parametric assumptions on the structure of the conditional survival function. We establish the convergence rate of our proposed estimator of the conditional survival function. In addition, we evaluate the performance of the proposed method through simulation studies and demonstrate its application on a real-world dataset. 1 Introduction Censored time-to-event data are widely encountered in various fields where understanding the timing of events, such as failure rates or disease progression, is critical, but the exact event times Correspondence author: Wenbin Lu, email: wlu4@ncsu.edu 1 arXiv:2503.09097v1 For example, estimating survival probability based on covariate information is essential for risk prediction, which plays a key role in developing and evaluating personalized medicine. The Kaplan-Meier (KM) estimator (Kaplan and Meier, 1958), Cox proportional hazards model (Cox, 1972), and random survival forests (Ishwaran et al., 2008) are commonly-used methods for estimating survival functions. The KM estimator is a non-parametric method suitable for population-level analyses. However, its utility is limited when the objective is to estimate conditional survival probabilities at the individual level. The Cox proportional hazards model offers a semi-parametric approach for estimating conditional survival functions, accommodating the incorporation of covariates.

As the use of machine learning models has increased, numerous studies have aimed to enhance fairness. However, research on the intersection of fairness and explainability remains insufficient, leading to potential issues in gaining the trust of actual users. Here, we propose a novel module that constructs a fair latent space, enabling faithful explanation while ensuring fairness. The fair latent space is constructed by disentangling and redistributing labels and sensitive attributes, allowing the generation of counterfactual explanations for each type of information. Our module is attached to a pretrained generative model, transforming its biased latent space into a fair latent space. Additionally, since only the module needs to be trained, there are advantages in terms of time and cost savings, without the need to train the entire generative model. We validate the fair latent space with various fairness metrics and demonstrate that our approach can effectively provide explanations for biased decisions and assurances of fairness.

Models trained with empirical risk minimization (ERM) are prone to be biased towards spurious correlations between target labels and bias attributes, which leads to poor performance on data groups lacking spurious correlations. It is particularly challenging to address this problem when access to bias labels is not permitted. To mitigate the effect of spurious correlations without bias labels, we first introduce a novel training objective designed to robustly enhance model performance across all data samples, irrespective of the presence of spurious correlations. From this objective, we then derive a debiasing method, Disagreement Probability based Resampling for debiasing (DPR), which does not require bias labels. DPR leverages the disagreement between the target label and the prediction of a biased model to identify bias-conflicting samples-those without spurious correlations-and upsamples them according to the disagreement probability. Empirical evaluations on multiple benchmarks demonstrate that DPR achieves state-of-the-art performance over existing baselines that do not use bias labels. Furthermore, we provide a theoretical analysis that details how DPR reduces dependency on spurious correlations.

This paper is the first to attempt differentially private (DP) topological data analysis (TDA), producing near-optimal private persistence diagrams. We analyze the sensitivity of persistence diagrams in terms of the bottleneck distance, and we show that the commonly used \v{C}ech complex has sensitivity that does not decrease as the sample size $n$ increases. This makes it challenging for the persistence diagrams of \v{C}ech complexes to be privatized. As an alternative, we show that the persistence diagram obtained by the $L^1$-distance to measure (DTM) has sensitivity $O(1/n)$. Based on the sensitivity analysis, we propose using the exponential mechanism whose utility function is defined in terms of the bottleneck distance of the $L^1$-DTM persistence diagrams. We also derive upper and lower bounds of the accuracy of our privacy mechanism; the obtained bounds indicate that the privacy error of our mechanism is near-optimal. We demonstrate the performance of our privatized persistence diagrams through simulations as well as on a real dataset tracking human movement.

The Generative Adversarial Network (GAN) was recently introduced in the literature as a novel machine learning method for training generative models. It has many applications in statistics such as nonparametric clustering and nonparametric conditional independence tests. However, training the GAN is notoriously difficult due to the issue of mode collapse, which refers to the lack of diversity among generated data. In this paper, we identify the reasons why the GAN suffers from this issue, and to address it, we propose a new formulation for the GAN based on randomized decision rules. In the new formulation, the discriminator converges to a fixed point while the generator converges to a distribution at the Nash equilibrium. We propose to train the GAN by an empirical Bayes-like method by treating the discriminator as a hyper-parameter of the posterior distribution of the generator. Specifically, we simulate generators from its posterior distribution conditioned on the discriminator using a stochastic gradient Markov chain Monte Carlo (MCMC) algorithm, and update the discriminator using stochastic gradient descent along with simulations of the generators. We establish convergence of the proposed method to the Nash equilibrium. Apart from image generation, we apply the proposed method to nonparametric clustering and nonparametric conditional independence tests. A portion of the numerical results is presented in the supplementary material.

Bayesian deep learning offers a principled way to address many issues concerning safety of artificial intelligence (AI), such as model uncertainty,model interpretability, and prediction bias. However, due to the lack of efficient Monte Carlo algorithms for sampling from the posterior of deep neural networks (DNNs), Bayesian deep learning has not yet powered our AI system. We propose a class of adaptive stochastic gradient Markov chain Monte Carlo (SGMCMC) algorithms, where the drift function is biased to enhance escape from saddle points and the bias is adaptively adjusted according to the gradient of past samples. We establish the convergence of the proposed algorithms under mild conditions, and demonstrate via numerical examples that the proposed algorithms can significantly outperform the existing SGMCMC algorithms, such as stochastic gradient Langevin dynamics (SGLD), stochastic gradient Hamiltonian Monte Carlo (SGHMC) and preconditioned SGLD, in both simulation and optimization tasks.