Package: c-lasso
c-lasso: a Python package for constrained sparse regression and classificationc-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 thedocumentation.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, sparse Huber M-estimation with linear equality constraints, and regularized Support Vector Machines.
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 asscikit-learnwhich CANNOT solve sparse constrained problems and general-purpose optimization solvers that do not scale well or are inaccurate (seebenchmarks) for the considered problems. In its current stage, however, c-lasso is not yet compatible with the scikit-learn API but rather a stand-alone tool.Below we show several use cases of the package, including an application of sparselog-contrastregression tasks forcompositionalmicrobiome data.The code builds on results from several papers which can be found in theReferences. We also refer to the accompanyingJOSS paper submission, also available onarXiv.Table of ContentsInstallationRegression and classification problemsGetting startedLog-contrast regression for microbiome dataOptimization schemesReferencesInstallationc-lasso is available on pip. You can install the package
in the shell usingpipinstallc-lassoTo use the c-lasso package in Python, typefromclassoimportclasso_problem# one can add auxiliary functions as well such as random_data or csv_to_npThec-lassopackage depends on the following Python packages:numpy;matplotlib;scipy;pandas;pytest(for tests)Regression and classification problemsThe 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] Constrained sparse Huber regression:This regression problem uses theHuber lossas objective function
for robust model fitting with l1 and linear equality constraints on the β vector. The parameter ρ=1.345.[R3] Constrained 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 (seeReferences[4,5] for further info).
This is the default problem formulation in c-lasso.[R4] Constrained 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 (seeReferences[4,5] for further info).[C1] Constrained sparse classification with Square Hinge loss:where the xiare 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] Constrained sparse classification with Huberized Square Hinge loss:where the xiare the rows of X and lρis 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 startedBasic exampleWe 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 routinerandom_datathat 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 inputzerosum=Trueimplies 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 = 50As we have not specified any problem, algorithm, or model selection settings, this problem instance
represents thedefaultsettings for a c-lasso instance:The problem is of regression type and uses formulation [R3], i.e. with concomitant scale estimation.Thedefaultoptimization scheme is the path algorithm (seeOptimization schemesfor further info).For model selection, stability selection at a theoretically derived λ value is used (seeReference[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 usingproblem.solve()After completion, the results of the optimization and model selection routines
can be visualized usingprint(problem.solution)The command shows the running time(s) for the c-lasso problem instance, and the selected variables for sability selectionSTABILITY SELECTION : 
   Selected variables :  7    63    148    164    168    
   Running time :  1.546sHere, we only used stability selection asdefaultmodel selection strategy.
The command also allows you to inspect the computed stability profile for all variables
at the theoretical λThe refitted β values on the selected support are also displayed in the next plotAdvanced exampleIn 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=Trueproblem.formulation.concomitant=Falseproblem.model_selection.CV=Trueproblem.model_selection.LAMfixed=Trueproblem.model_selection.PATH=Trueproblem.model_selection.StabSelparameters.method='max'problem.model_selection.CVparameters.seed=1problem.model_selection.LAMfixedparameters.rescaled_lam=Trueproblem.model_selection.LAMfixedparameters.lam=.1problem.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 = 0.09
          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    123    
     Running time :  0.104s

     PATH COMPUTATION : 
     Running time :  0.638s

     CROSS VALIDATION : 
     Selected variables :  16    17    57    59    64    73    74    76    93    115    123    134    137    181    
     Running time :  2.1s

     STABILITY SELECTION : 
     Selected variables :  17    59    76    123    137    
     Running time :  6.062sLog-contrast regression for microbiome dataIn thethe accompanying notebookwe study several microbiome data sets. We showcase two examples below.BMI prediction using the COMBO datasetWe first consider theCOMBO data setand show how to predict Body Mass Index (BMI) from microbial genus abundances and two non-compositional covariates  using "filtered_data".fromclassoimportcsv_to_np,classo_problem,clr# Load microbiome and covariate data XX0=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)# Load BMI measurements yy=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 datay=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 generaX=np.concatenate((X0,X_C,X_F,np.ones((len(X0),1))),axis=1)# Joint microbiome and covariate data and offsetlabel=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 problemproblem=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 R3problem.formulation.concomitant=Trueproblem.solve()print(problem)print(problem.solution)# Use formulation R4problem.formulation.huber=Trueproblem.formulation.concomitant=Trueproblem.solve()print(problem)print(problem.solution)pH prediction using the 88 soils datasetThe next microbiome example considers the88 soils datasetfromLauber et al., 2009.The task is to predict pH concentration in the soil from microbial abundance data. A similar analysis is available
inTree-Aggregated Predictive Modeling of Microbiome Datawith Central Park soil data fromRamirez et al..Code to run this application is available inthe accompanying notebookunderpH 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 = 0.3085The 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 :  0.287sOptimization schemesThe 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.References[1] B. R. Gaines, J. Kim, and H. Zhou,Algorithms for Fitting the Constrained Lasso, J. Comput. Graph. Stat., vol. 27, no. 4, pp. 861–871, 2018.[2] L. Briceno-Arias and S.L. Rivera,A Projected Primal–Dual Method for Solving Constrained Monotone Inclusions, J. Optim. Theory Appl., vol. 180, Issue 3, March 2019.[3] P. L. Combettes and J.C. Pesquet,Primal-Dual Splitting Algorithm for Solving Inclusions with Mixtures of Composite, Lipschitzian, and Parallel-Sum Type Monotone Operators, Set-Valued and Variational Analysis, vol. 20, pp. 307-330, 2012.[4] P. L. Combettes and C. L. Müller,Perspective M-estimation via proximal decomposition, Electronic Journal of Statistics, 2020,Journal version[5] P. L. Combettes and C. L. Müller,Regression models for compositional data: General log-contrast formulations, proximal optimization, and microbiome data applications, Statistics in Bioscience, 2020.[6] A. Mishra and C. L. Müller,Robust regression with compositional covariates, arXiv, 2019.[7] S. Rosset and J. Zhu,Piecewise linear regularized solution paths, Ann. Stat., vol. 35, no. 3, pp. 1012–1030, 2007.[8] J. Bien, X. Yan, L. Simpson, and C. L. Müller,Tree-Aggregated Predictive Modeling of Microbiome Data, biorxiv, 2020.