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.
Download