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Algorithms for constrained Lasso problems

# c-lasso: a Python package for constrained sparse regression and classification

c-lasso is a Python package that enables sparse and robust linear regression and classification with linear equality constraints on the model parameters. For detailed info, one can check the documentation.

The forward model is assumed to be:

Here, y and X are given outcome and predictor data. The vector y can be continuous (for regression) or binary (for classification). C is a general constraint matrix. The vector β comprises the unknown coefficients and σ an unknown scale.

The package handles several different estimators for inferring β (and σ), including the constrained Lasso, the constrained scaled Lasso, and sparse Huber M-estimation with linear equality constraints. Several different algorithmic strategies, including path and proximal splitting algorithms, are implemented to solve the underlying convex optimization problems.

We also include two model selection strategies for determining the sparsity of the model parameters: k-fold cross-validation and stability selection.

This package is intended to fill the gap between popular python tools such as scikit-learn which CANNOT solve sparse constrained problems and general-purpose optimization solvers that do not scale well for the considered problems.

Below we show several use cases of the package, including an application of sparse log-contrast regression tasks for compositional microbiome data.

The code builds on results from several papers which can be found in the References. We also refer to the accompanying JOSS paper submission, also available on arXiv.

## Installation

c-lasso is available on pip. You can install the package in the shell using

```pip install c-lasso
```

To use the c-lasso package in Python, type

```from classo import *
```

The `c-lasso` package depends on the following Python packages:

• `numpy`;
• `matplotlib`;
• `scipy`;
• `pandas`;
• `h5py`.

## Regression and classification problems

The c-lasso package can solve six different types of estimation problems: four regression-type and two classification-type formulations.

#### [R1] Standard constrained Lasso regression:

This is the standard Lasso problem with linear equality constraints on the β vector. The objective function combines Least-Squares for model fitting with l1 penalty for sparsity.

#### [R2] Contrained sparse Huber regression:

This regression problem uses the Huber loss as objective function for robust model fitting with l1 and linear equality constraints on the β vector. The parameter ρ=1.345.

#### [R3] Contrained scaled Lasso regression:

This formulation is similar to [R1] but allows for joint estimation of the (constrained) β vector and the standard deviation σ in a concomitant fashion (see References [4,5] for further info). This is the default problem formulation in c-lasso.

#### [R4] Contrained sparse Huber regression with concomitant scale estimation:

This formulation combines [R2] and [R3] to allow robust joint estimation of the (constrained) β vector and the scale σ in a concomitant fashion (see References [4,5] for further info).

#### [C1] Contrained sparse classification with Square Hinge loss:

where the xi are the rows of X and l is defined as:

This formulation is similar to [R1] but adapted for classification tasks using the Square Hinge loss with (constrained) sparse β vector estimation.

#### [C2] Contrained sparse classification with Huberized Square Hinge loss:

where the xi are the rows of X and lρ is defined as:

This formulation is similar to [C1] but uses the Huberized Square Hinge loss for robust classification with (constrained) sparse β vector estimation.

## Getting started

#### Basic example

We begin with a basic example that shows how to run c-lasso on synthetic data. This example and the next one can be found on the notebook 'Synthetic data Notebook.ipynb'

The c-lasso package includes the routine `random_data` that allows you to generate problem instances using normally distributed data.

```m, d, d_nonzero, k, sigma = 100, 200, 5, 1, 0.5
(X, C, y), sol = random_data(m, d, d_nonzero, k, sigma, zerosum=True, seed=1)
```

This code snippet generates a problem instance with sparse β in dimension d=100 (sparsity d_nonzero=5). The design matrix X comprises n=100 samples generated from an i.i.d standard normal distribution. The dimension of the constraint matrix C is d x k matrix. The noise level is σ=0.5. The input `zerosum=True` implies that C is the all-ones vector and Cβ=0. The n-dimensional outcome vector y and the regression vector β is then generated to satisfy the given constraints.

Next we can define a default c-lasso problem instance with the generated data:

```problem = classo_problem(X, y, C)
```

You can look at the generated problem instance by typing:

```print(problem)
```

This gives you a summary of the form:

``````FORMULATION: R3

MODEL SELECTION COMPUTED:
Stability selection

STABILITY SELECTION PARAMETERS:
numerical_method : not specified
method : first
B = 50
q = 10
percent_nS = 0.5
threshold = 0.7
lamin = 0.01
Nlam = 50
``````

As we have not specified any problem, algorithm, or model selection settings, this problem instance represents the default settings for a c-lasso instance:

• The problem is of regression type and uses formulation [R3], i.e. with concomitant scale estimation.
• The default optimization scheme is the path algorithm (see Optimization schemes for further info).
• For model selection, stability selection at a theoretically derived λ value is used (see Reference [4] for details). Stability selection comprises a relatively large number of parameters. For a description of the settings, we refer to the more advanced examples below and the API.

You can solve the corresponding c-lasso problem instance using

```problem.solve()
```

After completion, the results of the optimization and model selection routines can be visualized using

```print(problem.solution)
```

The command shows the running time(s) for the c-lasso problem instance, and the selected variables for sability selection

``````STABILITY SELECTION :
Selected variables :  1    5    14    17    18
Running time :  0.663s
``````

Here, we only used stability selection as default model selection strategy. The command also allows you to inspect the computed stability profile for all variables at the theoretical λ

The refitted β values on the selected support are also displayed in the next plot

In the next example, we show how one can specify different aspects of the problem formulation and model selection strategy.

```m,  d,  d_nonzero,  k, sigma = 100, 200, 5, 0, 0.5
(X, C, y), sol = random_data(m, d, d_nonzero, k, sigma, zerosum = True, seed = 4)
problem                                     = classo_problem(X, y, C)
problem.formulation.huber                   = True
problem.formulation.concomitant             = False
problem.model_selection.CV                  = True
problem.model_selection.LAMfixed            = True
problem.model_selection.PATH                = True
problem.model_selection.StabSelparameters.method = 'max'
problem.model_selection.CVparameters.seed = 1
problem.model_selection.LAMfixedparameters.rescaled_lam = True
problem.model_selection.LAMfixedparameters.lam = .1

problem.solve()
print(problem)

print(problem.solution)
```

Results :

``````FORMULATION: R2

MODEL SELECTION COMPUTED:
Lambda fixed
Path
Cross Validation
Stability selection

LAMBDA FIXED PARAMETERS:
numerical_method = Path-Alg
rescaled lam : True
threshold = 0.106
lam = 0.1
theoretical_lam = 0.224

PATH PARAMETERS:
numerical_method : Path-Alg
lamin = 0.001
Nlam = 80

CROSS VALIDATION PARAMETERS:
numerical_method : Path-Alg
one-SE method : True
Nsubset = 5
lamin = 0.001
Nlam = 80

STABILITY SELECTION PARAMETERS:
numerical_method : Path-Alg
method : max
B = 50
q = 10
percent_nS = 0.5
threshold = 0.7
lamin = 0.01
Nlam = 50

LAMBDA FIXED :
Selected variables :  17    59    76    123    137
Running time :  0.234s

PATH COMPUTATION :
Running time :  0.557s

CROSS VALIDATION :
Selected variables :  16    17    57    59    64    73    74    76    93    115    123    134    137    181
Running time :  1.751s

STABILITY SELECTION :
Selected variables :  1    3    7    12
Running time :  8.391s

``````

## Log-contrast regression for microbiome data

In the the accompanying notebook we study several microbiome data sets. We showcase two examples below.

#### BMI prediction using the COMBO dataset

We first consider the COMBO data set and show how to predict Body Mass Index (BMI) from microbial genus abundances and two non-compositional covariates using "filtered_data".

```from classo import *

# Load microbiome and covariate data X
X0  = csv_to_np('COMBO_data/complete_data/GeneraCounts.csv', begin = 0).astype(float)
X_C = csv_to_np('COMBO_data/CaloriData.csv', begin = 0).astype(float)
X_F = csv_to_np('COMBO_data/FatData.csv', begin = 0).astype(float)

y   = csv_to_np('COMBO_data/BMI.csv', begin = 0).astype(float)[:, 0]
labels = csv_to_np('COMBO_data/complete_data/GeneraPhylo.csv').astype(str)[:, -1]

# Normalize/transform data
y   = y - np.mean(y) #BMI data (n = 96)
X_C = X_C - np.mean(X_C, axis = 0)  #Covariate data (Calorie)
X_F = X_F - np.mean(X_F, axis = 0)  #Covariate data (Fat)
X0 = clr(X0, 1 / 2).T

# Set up design matrix and zero-sum constraints for 45 genera
X     = np.concatenate((X0, X_C, X_F, np.ones((len(X0), 1))), axis = 1) # Joint microbiome and covariate data and offset
label = np.concatenate([labels, np.array(['Calorie', 'Fat', 'Bias'])])
C = np.ones((1, len(X[0])))
C[0, -1], C[0, -2], C[0, -3] = 0., 0., 0.

# Set up c-lassso problem
problem = classo_problem(X, y, C, label = label)

# Use stability selection with theoretical lambda [Combettes & Müller, 2020b]
problem.model_selection.StabSelparameters.method      = 'lam'
problem.model_selection.StabSelparameters.threshold_label = 0.5

# Use formulation R3
problem.formulation.concomitant = True

problem.solve()
print(problem)
print(problem.solution)

# Use formulation R4
problem.formulation.huber = True
problem.formulation.concomitant = True

problem.solve()
print(problem)
print(problem.solution)
```

#### pH prediction using the Central Park soil dataset

The next microbiome example considers the Central Park Soil dataset from Ramirez et al.. The sample locations are shown in the Figure on the right. The task is to predict pH concentration in the soil from microbial abundance data. This task was also considered in Tree-Aggregated Predictive Modeling of Microbiome Data.

Code to run this application is available in the accompanying notebook under `pH data`. Below is a summary of a c-lasso problem instance (using the R3 formulation).

``````FORMULATION: R3

MODEL SELECTION COMPUTED:
Lambda fixed
Path
Stability selection

LAMBDA FIXED PARAMETERS:
numerical_method = Path-Alg
rescaled lam : True
threshold = 0.004
lam : theoretical
theoretical_lam = 0.2182

PATH PARAMETERS:
numerical_method : Path-Alg
lamin = 0.001
Nlam = 80

STABILITY SELECTION PARAMETERS:
numerical_method : Path-Alg
method : lam
B = 50
q = 10
percent_nS = 0.5
threshold = 0.7
lam = theoretical
theoretical_lam = 0.3085
``````

The c-lasso estimation results are summarized below:

``````LAMBDA FIXED :
Sigma  =  0.198
Selected variables :  14    18    19    39    43    57    62    85    93    94    104    107
Running time :  0.008s

PATH COMPUTATION :
Running time :  0.12s

STABILITY SELECTION :
Selected variables :  2    12    15
Running time :  0.287s
``````

## Optimization schemes

The available problem formulations [R1-C2] require different algorithmic strategies for efficiently solving the underlying optimization problem. We have implemented four algorithms (with provable convergence guarantees) that vary in generality and are not necessarily applicable to all problems. For each problem type, c-lasso has a default algorithm setting that proved to be the fastest in our numerical experiments.

### Path algorithms (Path-Alg)

This is the default algorithm for non-concomitant problems [R1,R3,C1,C2]. The algorithm uses the fact that the solution path along λ is piecewise- affine (as shown, e.g., in [1]). When Least-Squares is used as objective function, we derive a novel efficient procedure that allows us to also derive the solution for the concomitant problem [R2] along the path with little extra computational overhead.

### Projected primal-dual splitting method (P-PDS):

This algorithm is derived from [2] and belongs to the class of proximal splitting algorithms. It extends the classical Forward-Backward (FB) (aka proximal gradient descent) algorithm to handle an additional linear equality constraint via projection. In the absence of a linear constraint, the method reduces to FB. This method can solve problem [R1]. For the Huber problem [R3], P-PDS can solve the mean-shift formulation of the problem (see [6]).

### Projection-free primal-dual splitting method (PF-PDS):

This algorithm is a special case of an algorithm proposed in [3] (Eq.4.5) and also belongs to the class of proximal splitting algorithms. The algorithm does not require projection operators which may be beneficial when C has a more complex structure. In the absence of a linear constraint, the method reduces to the Forward-Backward-Forward scheme. This method can solve problem [R1]. For the Huber problem [R3], PF-PDS can solve the mean-shift formulation of the problem (see [6]).

### Douglas-Rachford-type splitting method (DR)

This algorithm is the most general algorithm and can solve all regression problems [R1-R4]. It is based on Doulgas Rachford splitting in a higher-dimensional product space. It makes use of the proximity operators of the perspective of the LS objective (see [4,5]) The Huber problem with concomitant scale [R4] is reformulated as scaled Lasso problem with the mean shift (see [6]) and thus solved in (n + d) dimensions.