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.

\[y \in \mathbb{R}^n, X = [x_1^\top, \dots, x_n^\top]^\top \in \mathbb{R}^{n \times p}\]

• Ordinary Least Squares (OLS):

\[\min_w \frac{1}{2} \|y - Xw\|^2_2\]

• Non-Negative Least Squares (NNLS):

\[\min_{w \geq 0} \frac{1}{2} \|y - Xw\|^2_2\]

• LASSO: L1-regularized least squares:

\[\min_w \frac{1}{2} \|y - Xw\|^2_2 + \lambda \|w\|_1\]

• L2-regularized logistic regression:

\[\min_w \sum_{i=1}^{n} \log(1 + \exp(-y_i x_i^\top w)) + \frac{\lambda}{2} \|w\|_2^2\]

• L1-regularized logistic regression:

\[\min_w \sum_{i=1}^{n} \log(1 + \exp(-y_i x_i^\top w)) + \lambda \|w\|_1\]

• L2-regularized Huber regression:

\[\min_{w, \sigma} {\sum_{i=1}^n \left(\sigma + H_{\epsilon}\left(\frac{X_{i}w - y_{i}}{\sigma}\right)\sigma\right) + \lambda {\|w\|_2}^2}\]

where

\[\begin{split}H_{\epsilon}(z) = \begin{cases} z^2, & \text {if } |z| < \epsilon, \\ 2\epsilon|z| - \epsilon^2, & \text{otherwise} \end{cases}\end{split}\]

• L1-regularized quantile regression:

\[\min_{w} \frac{1}{n} \sum_{i=1}^{n} PB_q(y_i - X_i w) + \lambda ||w||_1.\]

where \(PB_q\) is the pinball loss:

\[\begin{split}PB_q(t) = q \max(t, 0) + (1 - q) \max(-t, 0) = \begin{cases} q t, & t > 0, \\ 0, & t = 0, \\ (q - 1) t, & t < 0 \end{cases}\end{split}\]

• Linear ICA:

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

\[X = A S\]

where \(A \in \mathbb{R}^{d \times d}\) 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:

\[\min_B \frac{1}{2n} \sum_{i=1}^n \log |\textrm{diag} (B C^i B^{\top}) | - \log | B C^i B^{\top} |\]

where \(|\cdot|\) stands for the matrix determinant and \(\textrm{diag}\) 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.