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.

我们引入了一个在线数学框架，用于生存分析，允许对动态环境和被审查数据进行实时适应。通过优化性能高的二次在线凸优化算法 —— 在线牛顿阶段（ONS），我们能够估计事件时间分布。我们分析了 ONS 超参数的选择，该选择依赖于指数凹性质并对遗憾界限有重要影响。另外，我们提出了一种保证 ONS 对数随机遗憾的随机方法。最后，我们通过模拟实验来验证了这些论断。