Available benchmarks

Available benchmarks#

Note

Some benchmarks are briefly described in the list below. For a complete list of benchmarks, see GitHub repositories of the form benchopt/benchmark_*.

Notation: In what follows, $n$ (or n_samples) stands for the number of samples and $p ‘$ (or n_features) stands for the number of features.

y \in R^{n}, X = [x_{1}^{⊤}, \dots, x_{n}^{⊤}]^{⊤} \in R^{n \times p}

Ordinary Least Squares (OLS):

min_{w} \frac{1}{2} ‖ y - X w ‖_{2}^{2}

Non-Negative Least Squares (NNLS):

min_{w \geq 0} \frac{1}{2} ‖ y - X w ‖_{2}^{2}

LASSO: L1-regularized least squares:

min_{w} \frac{1}{2} ‖ y - X w ‖_{2}^{2} + λ ‖ w ‖_{1}

L2-regularized logistic regression:

min_{w} \sum_{i = 1}^{n} \log (1 + \exp (- y_{i} x_{i}^{⊤} w)) + \frac{λ}{2} ‖ w ‖_{2}^{2}

L1-regularized logistic regression:

min_{w} \sum_{i = 1}^{n} \log (1 + \exp (- y_{i} x_{i}^{⊤} w)) + λ ‖ w ‖_{1}

L2-regularized Huber regression:

min_{w, σ} \sum_{i = 1}^{n} (σ + H_{ϵ} (\frac{X_{i} w - y_{i}}{σ}) σ) + λ {‖ w ‖_{2}}^{2}

where

\begin{array}{r} H_{ϵ} (z) = {\begin{cases} z^{2}, & if | z | < ϵ, \\ 2 ϵ | z | - ϵ^{2}, & otherwise \end{cases} \end{array}

L1-regularized quantile regression:

min_{w} \frac{1}{n} \sum_{i = 1}^{n} P B_{q} (y_{i} - X_{i} w) + λ | | w | |_{1} .

where $P B_{q}$ is the pinball loss:

\begin{array}{r} P B_{q} (t) = q max (t, 0) + (1 - q) max (- t, 0) = {\begin{cases} q t, & t > 0, \\ 0, & t = 0, \\ (q - 1) t, & t < 0 \end{cases} \end{array}

Linear ICA:

Given some data $X \in R^{d \times n}$ assumed to be linearly related to unknown independent sources $S \in R^{d \times n}$ with

X = A S

where $A \in R^{d \times d}$ is also unknown, the objective of linear ICA is to recover $A$ up to permutation and scaling of its columns. The objective in this benchmark is related to some estimation on $A$ quantified with the so-called AMARI distance.

Approximate Joint Diagonalization (AJD):

Given n square symmetric positive matrices $C^{i}$ , it consists of solving the following problem:

min_{B} \frac{1}{2 n} \sum_{i = 1}^{n} \log | diag (B C^{i} B^{⊤}) | - \log | B C^{i} B^{⊤} |

where $| \cdot |$ stands for the matrix determinant and $diag$ stands for the operator that keeps only the diagonal elements of a matrix. Optionally, the matrix $B$ can be enforced to be orthogonal.

See benchmark_* repositories on GitHub for more.