## Introduction

Many advanced machine learning algorithms can be cast under the same framework that minimize the empirical risk ($$R_{emp}(w)$$) under the control of a regularization term ($$\Omega(w)$$): $\min_{w} J(w) := \lambda \Omega(w) + R_{emp}(w),$ where $R_{emp} := \frac{1}{m}\sum{l(x_i,y_i,w)}, \lambda > 0.$ @Teo_JMLR_2010 and @Do_JMLR_2012 have proposed efficient algorithms to solve this minimization problem. The bmrm package implements there solution together with several adapter functions that implement popular classification, regression and structure prediction algorithms. The adapter functions take the form of loss functions $$l(w,x_i,y_i)$$ that compute a loss value on each training example $$(x_i,y_i)$$. The algorithms currenty implemented are listed in the table below:

Learning Algorithm Description Type Loss Function Loss Value
Support Vector Machine (SVM) Linear Binary Classifier hingeLoss() $$\max(0,1-ywx)$$
Maximizing ROC area Linear Binary Classifier rocLoss() @Teo_JMLR_2010, §A.3.1
Maximizing fbeta score Linear Binary Classifier fbetaLoss() @Teo_JMLR_2010, §A.3.5
Logistic Regression Linear Binary Classifier logisticLoss() $$\log(1+e^{-ywx})$$
Least mean square regression Linear Regressor lmsRegressionLoss() $$(wx-y)^2/2$$
Least absolute deviation regression Linear Regressor ladRegressionLoss() $$abs(wx-y)$$
$$\epsilon$$-insensitive regression Linear Regressor epsilonInsensitiveRegressionLoss() $$\max(0,abs(wx-y)-\epsilon)$$
Quantile regression Linear Regressor quantileRegressionLoss() @Teo_JMLR_2010, Table 5
Multiclass SVM Structure Predictor softMarginVectorLoss() @Teo_JMLR_2010, Table 6
Ontology classification Structure Predictor ontologyLoss() @Teo_JMLR_2010, §A.4.2
Ordinal regression Structure Predictor ordinalRegressionLoss() @Teo_JMLR_2010, §A.3.2

Table: List of learning algorithms implemented in bmrm package. \label{tab:bmrm_learning_algorithms}

In addition to this implemented algorithms, the package is flexible enought to allow easy implementation of custom methods adapted to your learning problem.

Regarding regularization, bmrm package can handle both L1 and L2 regularization of the parameter vector $$w$$. L1 regularization is obtained by computing the L1-norm of the parameter vector ($$\Omega(w)=|w|$$); while L2 regularization is computed by using the L2-norm of the parameter vector ($$\Omega(w)=||w||$$). In theory, L1 regularization yield better model sparsity and may be prefered. However, the implementation available for L2-regularizer is much more memory efficient and can handle non-convex loss functions. The parameter $$\lambda$$ control the tradeoff between model fitting and model simplicity, and should be tuned to account for overfitting.

Most of the time, the loss functions are convex and all the ones implemented in the package are. However, non-convex losses can also be handle by the package if necessary. See section Choosing the Optimization Algorithms for more details.

## Quick start

In this quick start guide, we show how to train a multiclass-SVM on iris dataset with an intercept.

    library(bmrm)
x <- cbind(intercept=100,data.matrix(iris[c("Sepal.Length","Sepal.Width","Petal.Length","Petal.Width")]))
w <- nrbm(softMarginVectorLoss(x,iris$Species)) table(target=iris$Species,prediction=predict(w,x))


## Choosing the Optimization Algorithms

bmrm package implement the algorithm proposed by @Do_JMLR_2012 to solve the above minimization problem and a L1-regularized verion of the algorithm as well. The methods are respectively called nrbm() and nrbmL1(). nrbm() is memory optimized version and can handle non-convex risk when parameter convexRisk=FALSE. In contrast, nrbmL1() can handle L1-regularization, but doesn't support non-convex losses. In addition, by default nrbmL1() doesn't provide memory optimization to guaranty convergence of the algorithm. Memory optimization is however possible by setting parameter maxCP to a value typically below 100, but in this case convergence of the algorithm is not guaranty.

Regularizer Convex Loss Optimization method
L1 Yes nrbmL1()
L2 Yes or No nrbm()

Table: Recommended optimization method in the different use cases.

## Loss function

The loss functions has to accept a point estimate w (or 0) and return it with attributes lvalue and gradient set. lvalue contains the estimated loss value at the estimated point w; gradient contains the estimated gradient vector at the estimated point w.