Python implementation of elastic-net regularized generalized linear models

Pyglmnet is a Python library implementing generalized linear models (GLMs) with advanced regularization options. It provides a wide range of noise models (with paired canonical link functions) including gaussian, binomial, multinomial, poisson, and softplus. It supports a wide range of regularizers: ridge, lasso, elastic net, group lasso, and Tikhonov regularization.

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A brief introduction to GLMs

Linear models are estimated as

\[y = \beta_0 + X\beta + \epsilon\]

The parameters \(\beta_0, \beta\) are estimated using ordinary least squares, under the implicit assumption the \(y\) is normally distributed.

Generalized linear models allow us to generalize this approach to point-wise nonlinearities \(q(.)\) and a family of exponential distributions for \(y\).

\[y = q(\beta_0 + X\beta) + \epsilon\]

Regularized GLMs are estimated by minimizing a loss function specified by the penalized negative log-likelihood. The elastic net penalty interpolates between L2 and L1 norm. Thus, we solve the following optimization problem:

\[\min_{\beta_0, \beta} \frac{1}{N} \sum_{i = 1}^N \mathcal{L} (y_i, \beta_0 + \beta^T x_i) + \lambda [ \frac{1}{2}(1 - \alpha) \mathcal{P}_2 + \alpha \mathcal{P}_1 ]\]

where \(\mathcal{P}_2\) and \(\mathcal{P}_1\) are the generalized L2 (Tikhonov) and generalized L1 (Group Lasso) penalties, given by:

\[\begin{split}\mathcal{P}_2 & = & \|\Gamma \beta \|_2^2 \\ \mathcal{P}_1 & = & \sum_g \|\beta_{j,g}\|_2\end{split}\]

where \(\Gamma\) is the Tikhonov matrix: a square factorization of the inverse covariance matrix and \(\beta_{j,g}\) is the \(j\) th coefficient of group \(g\).

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