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FastSurvival: Hidden Computational Blessings in Training Cox Proportional Hazards Models
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Derives exact O(n) first and second partial derivatives (and third-order formula) of the Cox partial likelihood per coordinate, and builds monotone quadratic and cubic surrogate coordinate-descent optimizers used in a beam-search cardinality-constrained CPH solver.
Problem
Training the Cox proportional hazards model with Newton-type methods is computationally expensive and fails to converge globally (loss blow-up) or converges slowly at high precision, which hurts variable selection under highly correlated features.
Approach
Shows coordinate-wise first, second and third derivatives of the CPH loss are weighted central moments computable in O(n) via reverse cumulative sums, and uses the resulting Lipschitz bounds to build quadratic and cubic surrogate functions that are minimized coordinate-wise, ensuring monotone decrease; these are then plugged into a beam-search framework for cardinality-constrained CPH.
Key results
- Quadratic and cubic surrogate methods give monotonically decreasing loss, whereas Newton-type baselines blow up when regularization is weak (Flchain, lambda2=1).
- On synthetic data with rho=0.9 and 1200 samples, the proposed method is the only one to achieve 100% support recovery with a size-15 model (true support 15).
- On Dialysis, the method obtains the sparsest models (smallest number of coefficients) without losing test CIndex or IBS compared with other model classes.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use quadratic or cubic surrogate coordinate descent instead of Newton-type optimization | Cox proportional hazards training · Flchain, first fold | wall-clock training speed | Helped |
| Use quadratic or cubic surrogate coordinate descent instead of Newton-type optimization | Cox proportional hazards training · Flchain, first fold | loss behavior | Helped |
| Use FastSurvival beam search for feature selection | Cox proportional hazards models · Synthetic high-correlation data, 1200 samples | recovery rate: 100% | Helped |
| Use FastSurvival beam search for feature selection | Cox proportional hazards models · Employee Attrition | CIndex | Helped |
| Use FastSurvival beam search for feature selection | Cox proportional hazards models · Employee Attrition | IBS | Helped |
| Use FastSurvival for feature selection | Cox proportional hazards models · Dialysis | predictive performance at a given support size | Helped |
Surprising findings
- Newton-type CPH solvers can diverge or have loss blow-ups in early iterations, even though the CPH loss is convex.
- The central-moment connection does not extend to fourth and higher order derivatives (Lemma shows an extra term).
Limitations
- Proportional hazards and additive/linear model assumptions may not hold; the CPH model itself is not evaluated for assumption violations.
- Tree, forest, and neural survival models can capture nonlinear patterns that the CPH model cannot.
- Higher-order partial derivatives are not shown to be Lipschitz continuous; the central-moment pattern breaks for order four and above.
Experimental setup
Datasets: Flchain Employee Attrition Dialysis synthetic correlated datasets (rho=0.9)
Benchmarks: Flchain Employee Attrition Dialysis synthetic correlated datasets
Metrics: CPH loss CIndex IBS F1 score wall-clock time
Baselines: exact Newton quasi Newton proximal Newton Coxnet Abess Adaptive Lasso
Builds on: Cox proportional hazards model generalized orthogonal matching pursuit beam search for sparse learning proximal Newton quasi Newton
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