README.rst

Benchmark for Cox estimation

This benchmark is dedicated to solver of Cox estimation:

$$ \min_{w} \frac{1}{n} \sum_{i=1}^{n} -s_i \langle x_i, w \rangle + \log(\textstyle\sum_{y_j \geq y_i} e^{\langle x_j, w \rangle}) + \lambda \Big( \rho \lVert w \rVert_1 + \frac{1-\rho}{2} \lVert w \rVert^2_2 \Big) $$

where $n$ (or n_samples) stands for the number of samples, $p$ (or n_features) stands for the number of features, $s$ the vector of observation censorship, $y$ occurrences times.

$$\mathbf{X} \in \mathbb{R}^{n \times p} \ , \, s \in \{ 0, 1 \}^n, \ y \in \mathbb{R}^n, \quad w \in \mathbb{R}^p$$

In the case of tied data, data with observation having the same occurrences time, the objective reads

$$ \min_{w} \frac{1}{n} \sum_{l=1}^{m} \bigg( \sum_{i \in H_{i_l}} - \langle x_i, w \rangle + \log \Bigl(\textstyle \sum_{y_j \geq y_{i_l}} e^{\langle x_j, w \rangle} - \frac{\#(i) - 1}{\lvert H_{i_l} \rvert}\textstyle\sum_{j \in H_{i_l}} e^{\langle x_j, w \rangle}\Bigl) \bigg) + \lambda \Big( \rho \lVert w \rVert_1 + \frac{1-\rho}{2} \lVert w \rVert^2_2 \Big) $$

where $H_{i_l} = \{ i \ | \ s_i = 1 \ ;\ y_{i} = y_{i_l} \}$ is the set of uncensored observations with same occurrence time $y_{i_l}$ and $\#(i)$ the index of observation $i$ in $H_{i_l}$.

Install

This benchmark can be run using the following commands:

$ pip install -U benchopt
$ git clone https://github.com/benchopt/benchmark_l1_cox
$ cd benchmark_l1_cox
$ benchopt run .

Apart from the problem, options can be passed to benchopt run, to restrict the benchmarks to some solvers or datasets, e.g.:

$ benchopt run . -s solver1 -d dataset2 --max-runs 10 --n-repetitions 10

Use benchopt run -h for more details about these options, or visit https://benchopt.github.io/api.html.