11.2 Logistic regression¶

Logistic regression is an example of a binary classifier, where the output takes one two values 0 or 1 for each data point. We call the two values classes.

Formulation as an optimization problem

Define the sigmoid function

$S(x)=\frac{1}{1+\exp(-x)}.$

Next, given an observation $$x\in\real^d$$ and a weights $$\theta\in\real^d$$ we set

$h_\theta(x)=S(\theta^Tx)=\frac{1}{1+\exp(-\theta^Tx)}.$

The weights vector $$\theta$$ is part of the setup of the classifier. The expression $$h_\theta(x)$$ is interpreted as the probability that $$x$$ belongs to class 1. When asked to classify $$x$$ the returned answer is

$\begin{split}x\mapsto \begin{cases}\begin{array}{ll}1 & h_\theta(x)\geq 1/2, \\ 0 & h_\theta(x)<1/2.\end{array}\end{cases}\end{split}$

When training a logistic regression algorithm we are given a sequence of training examples $$x_i$$, each labelled with its class $$y_i\in \{0,1\}$$ and we seek to find the weights $$\theta$$ which maximize the likelihood function

$\prod_i h_\theta(x_i)^{y_i}(1-h_\theta(x_i))^{1-y_i}.$

Of course every single $$y_i$$ equals 0 or 1, so just one factor appears in the product for each training data point. By taking logarithms we can define the logistic loss function:

$J(\theta) = -\sum_{i:y_i=1} \log(h_\theta(x_i))-\sum_{i:y_i=0}\log(1-h_\theta(x_i)).$

The training problem with regularization (a standard technique to prevent overfitting) is now equivalent to

$\min_\theta J(\theta) + \lambda\|\theta\|_2.$

This can equivalently be phrased as

(11.22)$\begin{split}\begin{array}{lrllr} \minimize & \sum_i t_i +\lambda r & & & \\ \st & t_i & \geq - \log(h_\theta(x)) & = \log(1+\exp(-\theta^Tx_i)) & \mathrm{if}\ y_i=1, \\ & t_i & \geq - \log(1-h_\theta(x)) & = \log(1+\exp(\theta^Tx_i)) & \mathrm{if}\ y_i=0, \\ & r & \geq \|\theta\|_2. & & \end{array}\end{split}$

Implementation

As can be seen from (11.22) the key point is to implement the softplus bound $$t\geq \log(1+e^u)$$, which is the simplest example of a log-sum-exp constraint for two scalar variables $$t,u$$. This is equivalent to

$\exp(u-t) + \exp(-t)\leq 1$

and further to

(11.23)$\begin{split}\begin{array}{rclr} (z_1, 1, u-t) & \in & \EXP & (z_1\geq \exp(u-t)), \\ (z_2, 1, -t) & \in & \EXP & (z_2\geq \exp(-t)), \\ z_1+z_2 & \leq & 1. & \end{array}\end{split}$

To feed these constraints into MOSEK we add more auxiliary variables $$q_1,q_2,v_1,v_2$$ with constraints $$(z_1, q_1, v_1) \in \EXP$$, $$(z_2, q_2, v_2) \in \EXP$$, $$q_1=q_2=1$$, $$v_1=u-t$$ and $$v_2=-t$$.

Listing 11.10 Implementation of $$t\geq \log(1+e^u)$$ as in (11.23). Click here to download.
    // t >= log( 1 + exp(u) )
// t_i >= log( 1 + exp(u_i) ), i = 0..n-1
// Adds auxiliary variables and constraints
{
int z1 = nvar, z2 = nvar+n, v1 = nvar+2*n, v2 = nvar+3*n, q1 = nvar+4*n, q2 = nvar+5*n;
int zcon = ncon, v1con = ncon+n, v2con=ncon+2*n;
int[]    subi = new int[7*n];
int[]    subj = new int[7*n];
double[] aval = new double[7*n];
int         k = 0;

// Linear constraints
for(int i = 0; i < n; i++)
{
// z1 + z2 = 1
subi[k] = zcon+i;  subj[k] = z1+i;  aval[k] = 1;  k++;
subi[k] = zcon+i;  subj[k] = z2+i;  aval[k] = 1;  k++;
// u - t - v1 = 0
subi[k] = v1con+i; subj[k] = u+i;   aval[k] = 1;  k++;
subi[k] = v1con+i; subj[k] = t+i;   aval[k] = -1; k++;
subi[k] = v1con+i; subj[k] = v1+i;  aval[k] = -1; k++;
// - t - v2 = 0
subi[k] = v2con+i; subj[k] = t+i;   aval[k] = -1; k++;
subi[k] = v2con+i; subj[k] = v2+i;  aval[k] = -1; k++;
}

// Bounds for variables

// Cones
for(int i = 0; i < n; i++)
{
task.appendcone(conetype.pexp, 0.0, new int[]{z1+i, q1+i, v1+i});
task.appendcone(conetype.pexp, 0.0, new int[]{z2+i, q2+i, v2+i});
}
}


Once we have this subroutine, it is easy to implement a function that builds the regularized loss function model (11.22).

Listing 11.11 Implementation of (11.22). Click here to download.
    // Model logistic regression (regularized with full 2-norm of theta)
// X - n x d matrix of data points
// y - length n vector classifying training points
// lamb - regularization parameter
public static double[] logisticRegression(Env        env,
double[,]  X,
bool[]     y,
double     lamb)
{
int n = X.GetLength(0);
int d = X.GetLength(1);       // num samples, dimension

{
// Variables [r; theta; t; u]
int nvar = 1+d+2*n;
int r = 0, theta = 1, t = 1+d, u = 1+d+n;

// Constraints: theta'*X +/- u = 0

// Objective lambda*r + sum(t)
for(int i = 0; i < n; i++)

// The X block in theta'*X +/- u = 0
int[]    subi   = new int[d*n+n];
int[]    subj   = new int[d*n+n];
double[] aval   = new double[d*n+n];
int         k   = 0;
for(int i = 0; i < n; i++)
{
for(int j = 0; j < d; j++)
{
subi[k] = i; subj[k] = theta+j; aval[k] = X[i,j]; k++;
}
subi[d*n+i] = i; subj[d*n+i] = u+i;
if (y[i]) aval[d*n+i] = 1; else aval[d*n+i] = -1;
}

// Softplus function constraints

// Regularization

// Solution

In the next figure we apply logistic regression to the training set of 2D points taken from the example ex2data2.txt . The two-dimensional dataset was converted into a feature vector $$x\in\real^{28}$$ using monomial coordinates of degrees at most 6.