# RLScore: Regularized Least-Squares Learners

Uma boa alternativa para ensemble quando a dimensionalidade dos datasets for alta, ou as alternativas com Elastic Net, Lasso e Ridge não derem a convergência desejada.

RLScore: Regularized Least-Squares Learners

RLScore is a Python open source module for kernel based machine learning. The library provides implementations of several regularized least-squares (RLS) type of learners. RLS methods for regression and classification, ranking, greedy feature selection, multi-task and zero-shot learning, and unsupervised classification are included. Matrix algebra based computational short-cuts are used to ensure efficiency of both training and cross-validation. A simple API and extensive tutorials allow for easy use of RLScore.

Regularized least squares (RLS) is a family of methods for solving the least-squares problem while using regularization to further constrain the resulting solution. RLS is used for two main reasons. The first comes up when the number of variables in the linear system exceeds the number of observations. In such settings, the ordinary least-squares problem is ill-posed and is therefore impossible to fit because the associated optimization problem has infinitely many solutions. RLS allows the introduction of further constraints that uniquely determine the solution. The second reason that RLS is used occurs when the number of variables does not exceed the number of observations, but the learned model suffers from poor generalization. RLS can be used in such cases to improve the generalizability of the model by constraining it at training time. This constraint can either force the solution to be “sparse” in some way or to reflect other prior knowledge about the problem such as information about correlations between features. A Bayesian understanding of this can be reached by showing that RLS methods are often equivalent to priors on the solution to the least-squares problem. To sse in Depth Installation 1) `\$ pip install rlscore` 2) `\$ export CFLAGS="-I /usr/local/lib/python2.7/site-packages/numpy/core/include \$CFLAGS"` Original post

In :

# Import libraries import numpy as np from rlscore.learner import RLS from rlscore.measure import sqerror from rlscore.learner import LeaveOneOutRLS

In :

# Function to load dataset and split in train and test sets def load_housing(): np.random.seed(1) D = np.loadtxt(“/Volumes/PANZER/Github/learning-space/Datasets/02 - Classification/housing_data.txt”) np.random.shuffle(D) X = D[:,:-1] # Independent variables Y = D[:,-1] # Dependent variable X_train = X[:250] Y_train = Y[:250] X_test = X[250:] Y_test = Y[250:] return X_train, Y_train, X_test, Y_test

In :

def print_stats(): X_train, Y_train, X_test, Y_test = load_housing() print(“Housing data set characteristics”) print(“Training set: %d instances, %d features” %X_train.shape) print(“Test set: %d instances, %d features” %X_test.shape) if __name__ == “__main__”: print_stats()

Housing data set characteristics Training set: 250 instances, 13 features Test set: 256 instances, 13 features

# Linear regression with default parameters

In :

# Function to train RLS method def train_rls(): #Trains RLS with default parameters (regparam=1.0, kernel=’LinearKernel’) X_train, Y_train, X_test, Y_test = load_housing() learner = RLS(X_train, Y_train) #Leave-one-out cross-validation predictions, this is fast due to #computational short-cut P_loo = learner.leave_one_out() #Test set predictions P_test = learner.predict(X_test) # Stats print(“leave-one-out error %f” %sqerror(Y_train, P_loo)) print(“test error %f” %sqerror(Y_test, P_test)) #Sanity check, can we do better than predicting mean of training labels? print(“mean predictor %f” %sqerror(Y_test, np.ones(Y_test.shape)*np.mean(Y_train))) if __name__==”__main__”: train_rls()

leave-one-out error 25.959399 test error 25.497222 mean predictor 81.458770

# Choosing regularization parameter with leave-one-out

Regularization parameter with grid search in exponential grid to catch the lowest LOO-CV error.

In :

def train_rls(): #Select regparam with leave-one-out cross-validation X_train, Y_train, X_test, Y_test = load_housing() learner = RLS(X_train, Y_train) best_regparam = None best_error = float(“inf”) #exponential grid of possible regparam values log_regparams = range(-15, 16) for log_regparam in log_regparams: regparam = 2.**log_regparam #RLS is re-trained with the new regparam, this #is very fast due to computational short-cut learner.solve(regparam) #Leave-one-out cross-validation predictions, this is fast due to #computational short-cut P_loo = learner.leave_one_out() e = sqerror(Y_train, P_loo) print(“regparam 2**%d, loo-error %f” %(log_regparam, e)) if e < best_error: best_error = e best_regparam = regparam learner.solve(best_regparam) P_test = learner.predict(X_test) print(“best regparam %f with loo-error %f” %(best_regparam, best_error)) print(“test error %f” %sqerror(Y_test, P_test)) if __name__==”__main__”: train_rls()

regparam 2**-15, loo-error 24.745479 regparam 2**-14, loo-error 24.745463 regparam 2**-13, loo-error 24.745431 regparam 2**-12, loo-error 24.745369 regparam 2**-11, loo-error 24.745246 regparam 2**-10, loo-error 24.745010 regparam 2**-9, loo-error 24.744576 regparam 2**-8, loo-error 24.743856 regparam 2**-7, loo-error 24.742982 regparam 2**-6, loo-error 24.743309 regparam 2**-5, loo-error 24.750966 regparam 2**-4, loo-error 24.786243 regparam 2**-3, loo-error 24.896991 regparam 2**-2, loo-error 25.146493 regparam 2**-1, loo-error 25.537315 regparam 2**0, loo-error 25.959399 regparam 2**1, loo-error 26.285436 regparam 2**2, loo-error 26.479254 regparam 2**3, loo-error 26.603001 regparam 2**4, loo-error 26.801196 regparam 2**5, loo-error 27.352322 regparam 2**6, loo-error 28.837002 regparam 2**7, loo-error 32.113350 regparam 2**8, loo-error 37.480625 regparam 2**9, loo-error 43.843555 regparam 2**10, loo-error 49.748687 regparam 2**11, loo-error 54.912297 regparam 2**12, loo-error 59.936226 regparam 2**13, loo-error 65.137825 regparam 2**14, loo-error 70.126118 regparam 2**15, loo-error 74.336978 best regparam 0.007812 with loo-error 24.742982 test error 24.509981

# Training with RLS and simultaneously selecting the regularization parameter with leave-one-out using LeaveOneOutRLS

In :

def train_rls(): #Trains RLS with automatically selected regularization parameter X_train, Y_train, X_test, Y_test = load_housing() # Grid search regparams = [2.**i for i in range(-15, 16)] learner = LeaveOneOutRLS(X_train, Y_train, regparams = regparams) loo_errors = learner.cv_performances P_test = learner.predict(X_test) print(“leave-one-out errors “ +str(loo_errors)) print(“chosen regparam %f” %learner.regparam) print(“test error %f” %sqerror(Y_test, P_test)) if __name__==”__main__”: train_rls()

leave-one-out errors [ 24.74547881 24.74546295 24.74543138 24.74536884 24.74524616 24.74501033 24.7445764 24.74385625 24.74298177 24.74330936 24.75096639 24.78624255 24.89699067 25.14649266 25.53731465 25.95939943 26.28543584 26.47925431 26.6030015 26.80119588 27.35232186 28.83700156 32.11334986 37.48062503 43.84355496 49.7486873 54.91229746 59.93622566 65.1378248 70.12611801 74.33697809] chosen regparam 0.007812 test error 24.509981

# Learning nonlinear predictors using kernels

RLS using a non-linear kernel function.

In :

def train_rls(): #Selects both the gamma parameter for Gaussian kernel, and regparam with loocv X_train, Y_train, X_test, Y_test = load_housing() regparams = [2.**i for i in range(-15, 16)] gammas = regparams best_regparam = None best_gamma = None best_error = float(“inf”) for gamma in gammas: #New RLS is initialized for each kernel parameter learner = RLS(X_train, Y_train, kernel=“GaussianKernel”, gamma=gamma) for regparam in regparams: #RLS is re-trained with the new regparam, this #is very fast due to computational short-cut learner.solve(regparam) #Leave-one-out cross-validation predictions, this is fast due to #computational short-cut P_loo = learner.leave_one_out() e = sqerror(Y_train, P_loo) #print “regparam”, regparam, “gamma”, gamma, “loo-error”, e if e < best_error: best_error = e best_regparam = regparam best_gamma = gamma learner = RLS(X_train, Y_train, regparam = best_regparam, kernel=“GaussianKernel”, gamma=best_gamma) P_test = learner.predict(X_test) print(“best parameters gamma %f regparam %f” %(best_gamma, best_regparam)) print(“best leave-one-out error %f” %best_error) print(“test error %f” %sqerror(Y_test, P_test)) if __name__==”__main__”: train_rls()

best parameters gamma 0.000031 regparam 0.000244 best leave-one-out error 21.910837 test error 16.340877

# Binary classification and Area under ROC curve

In :

from rlscore.utilities.reader import read_svmlight # Load dataset and stats def print_stats(): X_train, Y_train, foo = read_svmlight(“/Volumes/PANZER/Github/learning-space/Datasets/02 - Classification/a1a.t”) X_test, Y_test, foo = read_svmlight(“/Volumes/PANZER/Github/learning-space/Datasets/02 - Classification/a1a”) print(“Adult data set characteristics”) print(“Training set: %d instances, %d features” %X_train.shape) print(“Test set: %d instances, %d features” %X_test.shape) if __name__==”__main__”: print_stats()

Adult data set characteristics Training set: 30956 instances, 123 features Test set: 1605 instances, 119 features

In [ ]:

from rlscore.learner import RLS from rlscore.measure import accuracy from rlscore.utilities.reader import read_svmlight def train_rls(): # Train ans test datasets X_train, Y_train, foo = read_svmlight(“/Volumes/PANZER/Github/learning-space/Datasets/02 - Classification/a1a.t”) X_test, Y_test, foo = read_svmlight(“/Volumes/PANZER/Github/learning-space/Datasets/02 - Classification/a1a”, X_train.shape) learner = RLS(X_train, Y_train) best_regparam = None best_accuracy = 0. #exponential grid of possible regparam values log_regparams = range(-15, 16) for log_regparam in log_regparams: regparam = 2.**log_regparam #RLS is re-trained with the new regparam, this #is very fast due to computational short-cut learner.solve(regparam) #Leave-one-out cross-validation predictions, this is fast due to #computational short-cut P_loo = learner.leave_one_out() acc = accuracy(Y_train, P_loo) print(“regparam 2**%d, loo-accuracy %f” %(log_regparam, acc)) if acc > best_accuracy: best_accuracy = acc best_regparam = regparam learner.solve(best_regparam) P_test = learner.predict(X_test) print(“best regparam %f with loo-accuracy %f” %(best_regparam, best_accuracy)) print(“test set accuracy %f” %accuracy(Y_test, P_test)) if __name__==”__main__”: train_rls()