Cox Regression Learns From Truncated Data Orders of Magnitude Faster Than EM

A group of researchers has published arXiv preprint 2610.12538 on survival models trained on data filtered by the outcome itself. On four datasets, accounting for truncation reduces the prediction bias of standard Cox regression, and the new approach outpaces the standard expectation-maximization (EM) algorithm by orders of magnitude. The main theoretical result: when the baseline hazard is known, the Cox model can be trained with ordinary first-order optimization, even though the loss function is non-convex.

A sample that distorts the picture

Most models are trained on the assumption that the training data resembles what the model will encounter in practice. Real datasets often fail this condition. The authors take one particular case: an observation enters the dataset depending on its label. This is called outcome-dependent truncation. A generic example: a cohort includes only people who survived until they were enrolled in the study. Short trajectories are simply missing from the data.

Right censoring comes on top of truncation. For some subjects, the event has not occurred by the end of observation, and all that is known is that it will happen after a certain point. Survival analysis has handled censoring for a long time. The literature on efficient learning under truncation developed separately. This work brings the two directions together.

What is proven and where the trap is hidden

Existing methods for truncated survival data did not describe the geometry of the optimization problem. They relied on iterative procedures such as EM, which work but are slow and offer no guarantees about the loss landscape. The authors close this gap.

The result for a known baseline hazard: Cox regression on truncated and right-censored data can be learned efficiently with first-order methods. Nonlinearity does not get in the way, even though the problem is non-convex.

If the baseline hazard is estimated jointly with the other parameters, the picture changes. The researchers construct models that are identifiable and correctly specified, yet their negative log-likelihood has strict suboptimal local minima. A gradient method will stop at such a point even though the true solution lies elsewhere. The practical takeaway: a correctly posed problem does not, by itself, guarantee that the optimizer will reach the answer.

What the experiments showed

The evaluation used four survival datasets. The authors compared naive Cox regression, which ignores truncation, with a variant that accounts for it.

• Naive Cox regression produces biased predictions when the data is truncated by outcome.
• Accounting for truncation reduces this bias on all four datasets.
• The approach carries over from the linear model to neural networks.
• A speedup of orders of magnitude over standard EM is claimed.
• Convergence guarantees are proven for a known baseline hazard; for joint learning, the paper shows counterexamples with local minima.

Where this is useful

The preprint has not yet been peer reviewed, and the abstract names no datasets and gives no exact bias values. The size of the effect is therefore best checked in the full text and in the published code, if it appears.

The takeaway for practitioners is this. If your sample is formed by a condition tied to the outcome, for example in medical registries, insurance data, customer churn, or equipment lifetime estimation, a naive Cox model will err systematically. When the shape of the baseline hazard can be taken from external sources, it is worth fixing it and using gradient optimization with guarantees. If the baseline hazard has to be learned together with the model, run the optimization from several starting points and compare the results. For teams where EM has been a bottleneck on large samples, an orders-of-magnitude speedup means they can retrain models more often and try more architectures.