  In many biomedical applications, outcome is measured as a ``time-to-event''
(eg. disease progression or death). To assess the connection between features
of a patient and this outcome, it is common to assume a proportional hazards
model, and fit a proportional hazards regression (or Cox regression). To fit
this model, a log-concave objective function known as the ``partial
likelihood'' is maximized. For moderate-sized datasets, an efficient
Newton-Raphson algorithm that leverages the structure of the objective can be
employed. However, in large datasets this approach has two issues: 1) The
computational tricks that leverage structure can also lead to computational
instability; 2) The objective does not naturally decouple: Thus, if the dataset
does not fit in memory, the model can be very computationally expensive to fit.
This additionally means that the objective is not directly amenable to
stochastic gradient-based optimization methods. To overcome these issues, we
propose a simple, new framing of proportional hazards regression: This results
in an objective function that is amenable to stochastic gradient descent. We
show that this simple modification allows us to efficiently fit survival models
with very large datasets. This also facilitates training complex, eg.
neural-network-based, models with survival data.
