ENH - Implement Cox with Efron estimate

[Badr-MOUFAD]

A follow-up of #157,

Handling tied data using Efron estimate can be obtained by slightly modifying the Cox datafit as follows

$$ l(\beta) = -\langle s, \mathbf{X}\beta \rangle + \langle s, \log(\mathbf{B}e^{\mathbf{X}\beta} - \mathbf{A}e^{\mathbf{X}\beta}) \rangle $$

where $\mathbf{A}$ is a matrix chosen accordingly to account for the additional term in the Breslow $\log$.
Also, evaluating $\mathbf{A} v$ or $\mathbf{A}^\top v$ is cheap and can be obtained in linear time.

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Link to the maths behind

[mathurinm]

Maybe you can add a small section at the end of the doc example with lifelines to show that we handle ties the same way lifelines does? no need for another speed benchmark, just showing that we have this functionality too

[Badr-MOUFAD]

Maybe you can add a small section at the end of the doc example with lifelines to show that we handle ties the same way lifelines does? no need for another speed benchmark, just showing that we have this functionality too

Yes, I agree with the idea!

[mathurinm]

The edited example and benchmark in this PR may interest @ogrisel too :)

[ogrisel]

Haha the benchmark plot! 😅

[ogrisel]

BTW do you see a similar perf improvement for smooth penalties?

[Badr-MOUFAD]

@ogrisel we have added support for L2 regularization too (with scipy LBFGS and our Prox Newton methods).
Check the bench figure

Also, here is the link to the complete benchmark and the repo to reproduce it.