We introduce an online mathematical framework for survival analysis, allowing real time adaptation to dynamic environments and censored data. This framework enables the estimation of event time distributions through an optimal second order online convex optimization algorithm-Online Newton Step (ONS). This approach, previously unexplored, presents substantial advantages, including explicit algorithms with non-asymptotic convergence guarantees. Moreover, we analyze the selection of ONS hyperparameters, which depends on the exp-concavity property and has a significant influence on the regret bound. We propose a stochastic approach that guarantees logarithmic stochastic regret for ONS. Additionally, we introduce an adaptive aggregation method that ensures robustness in hyperparameter selection while maintaining fast regret bounds. The findings of this paper can extend beyond the survival analysis field, and are relevant for any case characterized by poor exp-concavity and unstable ONS. Finally, these assertions are illustrated by simulation experiments.

翻译：我们提出了一种用于生存分析的在线数学框架，该框架支持实时适应动态环境与删失数据。该框架通过最优二阶在线凸优化算法——在线牛顿步（ONS）实现事件时间分布的估计。这一此前未被探索的方法展现出显著优势，包括提供具有非渐近收敛保证的显式算法。此外，我们分析了ONS超参数的选择问题，该选择取决于指数凹性（exp-concavity）特性并对遗憾界（regret bound）有重要影响。我们提出了一种可确保ONS对数随机遗憾的随机方法。同时，我们引入了一种自适应聚合方法，该方法在保持快速遗憾界的同时保证超参数选择的鲁棒性。本文的研究发现可超越生存分析领域，适用于任何具有弱指数凹性与ONS不稳定性特征的场景。最后，通过仿真实验验证了上述论断。