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Online Learning Approach for Survival Analysis

Camila Fernandez, Pierre Gaillard, Joseph de Vilmarest, Olivier Wintenberger

TL;DR

This work casts online survival analysis as an online convex optimization problem using an exponential-hazard model and the Online Newton Step (ONS) to enable real-time updating of event-time distributions under censoring. It analyzes hyperparameter sensitivity via exp-concavity and directional-derivative conditions, and introduces a stochastic variant with logarithmic regret guarantees. To address worst-case learning-rate behavior, the paper proposes SurvONS, an aggregation scheme based on Bernstein Online Aggregation (BOA) that adaptively combines multiple ONS instances with different learning rates. Simulation demonstrates the method's robustness and suggests that larger calibration grids improve performance, with the approach potentially extending to other domains characterized by weak exp-concavity.

Abstract

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.

Online Learning Approach for Survival Analysis

TL;DR

This work casts online survival analysis as an online convex optimization problem using an exponential-hazard model and the Online Newton Step (ONS) to enable real-time updating of event-time distributions under censoring. It analyzes hyperparameter sensitivity via exp-concavity and directional-derivative conditions, and introduces a stochastic variant with logarithmic regret guarantees. To address worst-case learning-rate behavior, the paper proposes SurvONS, an aggregation scheme based on Bernstein Online Aggregation (BOA) that adaptively combines multiple ONS instances with different learning rates. Simulation demonstrates the method's robustness and suggests that larger calibration grids improve performance, with the approach potentially extending to other domains characterized by weak exp-concavity.

Abstract

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.
Paper Structure (34 sections, 14 theorems, 153 equations, 4 figures, 1 table, 2 algorithms)

This paper contains 34 sections, 14 theorems, 153 equations, 4 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background on parametric inference
  3. Notation
  4. Survival probability
  5. Likelihood
  6. Sequential likelihood optimization
  7. Online convex optimization
  8. Setting
  9. Exp-concavity and directional derivative condition
  10. Online Newton Step
  11. Stochastic setting
  12. Stochastic Model
  13. Stochastically Exp-Concavity
  14. Stochastic Regret
  15. Survival ONS algorithm
  16. ...and 19 more sections

Key Result

Proposition 1

Under the exponential model from Definition ass:exp and omitting additional constants, the negative log-likelihood function $\ell :\Theta \rightarrow \mathbb R$ can be written in the following way:

Figures (4)

  • Figure 1: Regret bound orders (up to logarithmic factors)
  • Figure 2: Density of $\gamma_t$ estimation obtained by Algorithm \ref{['alg:BOAm']}, with $\Gamma_1$ [left] and $\Gamma_2$ [right]
  • Figure 3: Cumulative negative log-likelihood with hyperparameters in grid $\Gamma_1$ [left] and $\Gamma_2$ [right]
  • Figure 4: Average quadratic error with hyperparameters in $\Gamma_1$ [left] and $\Gamma_2$ [right]

Theorems & Definitions (29)

  • Definition 1: Log-linear regression model for the Hazard function
  • Proposition 1
  • Definition 2
  • Definition 3
  • Lemma 1
  • Lemma 2
  • Theorem 1: Hazan hazan2022introduction
  • Definition 4: Stochastic exp-concavity
  • Theorem 2
  • Corollary 1
  • ...and 19 more