# SGD for Cox proportional hazards model

I am wondering if there's any implementation or theory about using stochastic gradient descent for estimation parameters in Cox proportional hazards model. Normally in that model the partial-log-likelihood (which for right censoring that is independent and uninformative about parameters is a true log-likelihood)

$$\ell_p(\beta) = \sum\limits_{i=1}^{K}X_i'\beta - \sum\limits_{i=1}^{K}\log\Big(\sum\limits_{l\in \mathscr{R}(t_i)}^{}e^{X_l'\beta}\Big)$$

where

• $K$ denotes number of failure times,
• $\mathscr{R}(t_i)$ is a risk set at time $t_i$

is estimated using the first and second derivatives and Newton-Raphson algorithm.

I was wondering that if the first derivative can be formulated as a sum of elements in which every element corresponds only to information from 1 observation, then the estimation with the stochastic gradient descent would be easy, but I can't transform this derivative to such form:

$$U_k(\beta)=\dfrac{\partial\ell_k(\beta)}{\partial\beta_k}=\sum\limits_{i=1}^{K}\Big(X_{ik}-A_{ik}\Big), A_{ik} = \dfrac{\sum X_{lk} e^{X_l'\beta}}{\sum e^{X_l'\beta}}$$ and the sums in $A_{ik}$ are over risk set.

Does anyone heard about any articles applying SGD to Cox proportional hazards model? It might have great applications in large-scale cox models.

• Thanks for your comment. Do you rather suggest to reorder samples and then go one by one to estimate coefficients of the cox model by using previously observed samples? So after reordering, for an n-th observation I calculate only 1 factor from the score function based on n-1 previously observed samples and that n-th one AND then actualize coefficients? Or rather I reorder samples, and then go by let's say 5 sample and I only use 5 samples to calculate coefficents based on those observed 5 examples and whole factors from the score function that appear to exist for those 5 observations? – Marcin Kosiński Sep 8 '15 at 20:18