PyTorch implementation of group elastic net

# torch-gel

This package provides PyTorch implementations to solve the group elastic net problem. Let Aj (j = 1 … p) be feature matrices of sizes m × nj (m is the number of samples, and nj is the number of features in the jth group), and let y be an m × 1 vector of the responses. Group elastic net finds coefficients βj, and a bias β0 that solve the optimization problem

min β0, …, βp ½ ║y - β0 - ∑ Aj βj║2 + m ∑ √nj (λ1║βj║ + λ2║βj║2).

Here λ1 and λ2 are scalar coefficients that control the amount of 2-norm and squared 2-norm regularization. This 2-norm regularization encourages sparsity at the group level; entire βj might become 0. The squared 2-norm regularization is in similar spirit to elastic net, and addresses some of the issues of lasso. Note that group elastic net includes as special cases group lasso (λ2 = 0), ridge regression (λ1 = 0), elastic net (each nj = 1), and lasso (each nj = 1 and λ2 = 0). The optimization problem is convex, and can be solved efficiently. This package provides two implementations; one based on proximal gradient descent, and one based on coordinate descent.

## Installation

Install with `pip`

```pip install torchgel
```

`tqdm` (for progress bars), and numpy are pulled in as dependencies. PyTorch (`v1.0+`) is also needed, and needs to be installed manually. Refer to the PyTorch website for instructions.

## Usage

`examples/main.ipynb` is a Jupyter notebook that walks through using the package for a typical use-case. A more formal description of the functions follows; and for details about the algorithms, refer to the docstrings of files in the `gel` directory.

### Solving Single Instances

The modules `gel.gelfista` and `gel.gelcd` provide implementations based on proximal gradient descent and coordinate descent respectively. Both have similar interfaces, and expose two main public functions: `make_A` and `gel_solve`. The feature matrices should be stored in a list (say `As`) as PyTorch tensor matrices, and the responses should be stored in a PyTorch vector (say `y`). Additionally, the sizes of the groups (nj) should be stored in a vector (say `ns`). First use the `make_A` function to convert the feature matrices into a suitable format:

```A = make_A(As, ns)
```

Then pass `A`, `y` and other required arguments to `gel_solve`. The general interface is::

```b_0, B = gel_solve(A, y, l_1, l_2, ns, **kwargs)
```

`l_1` and `l_2` are floats representing λ1 and λ2 respectively. The method returns a float `b_0` representing the bias and a PyTorch matrix `B` holding the other coefficients. `B` has size p × maxj nj with suitable zero padding. The following sections cover additional details for the specific implementations.

The `gel.gelfista` module contains a proximal gradient descent implementation. It's usage is just as described in the template above. Refer to the docstring for `gel.gelfista.gel_solve` for details about the other arguments.

#### Coordinate Descent

The `gel.gelcd` module contains a coordinate descent implementation. Its usage is a bit more involved than the FISTA implementation. Coordinate descent iteratively solves single blocks (each corresponding to a single βj). There are multiple solvers provided to solve the individual blocks. These are the `gel.gelcd.block_solve_*` functions. Refer to their docstrings for details about their arguments. `gel.gelcd.gel_solve` requires passing a block solve function and its arguments (as a dictionary). Refer to its docstring for further details.

### Solution Paths

`gel.gelpaths` provides a wrapper function `gel_paths` to solve the group elastic net problem for multiple values of the regularization coefficients. It implements a two-stage process. For a given λ1 and λ2, first the group elastic net problem is solved and the feature blocks with non-zero coefficients is extracted (the support). Then ridge regression models are learned for each of several provided regularization values. The final model is summarized using an arbitrary provided summary function, and the summary for each combination of the regularization values is returned as a dictionary. The docstring contains more details. `gel.ridgepaths` contains another useful function, `ridge_paths` which can efficiently solve ridge regression for multiple regularization values.

## Citation

``````@misc{koushik2017torchgel,
author = {Koushik, Jayanth},
title = {torch-gel},
year = {2017},
publisher = {GitHub},
journal = {GitHub repository},
howpublished = {\url{https://github.com/jayanthkoushik/torch-gel}},
}
``````

## Project details

This version 2.0.0 1.0.0 0.22.0 0.21.0 0.10.0 0.9.0 0.8.2 0.8.1.post1 0.8.1 0.8.0 0.7.6 0.7.5 0.7.4 0.7.3 0.7.2 0.7.1 0.7.0 0.6.5 0.6.4 0.6.3 0.6.2 0.6.1