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

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

Implementation

As can be seen from (11.13) 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 terms. Here \(t\) is a scalar variable and \(u\) will be the affine expression of the form \(\pm \theta^Tx_i\). This is equivalent to

\[\exp(u-t) + \exp(-t)\leq 1\]

and further to

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

This formulation can be entered using affine conic constraints (see Sec. 6.2 (From Linear to Conic Optimization)).

Listing 11.7 Implementation of \(t\geq \log(1+e^u)\) as in (11.14). Click here to download.
# Adds ACCs for t_i >= log ( 1 + exp((1-2*y[i]) * theta' * X[i]) )
# Adds auxiliary variables, AFE rows and constraints
def softplus(task, d, n, theta, t, X, y):
    nvar = task.getnumvar()
    ncon = task.getnumcon()
    nafe = task.getnumafe()
    task.appendvars(2*n)    # z1, z2
    task.appendcons(n)      # z1 + z2 = 1
    task.appendafes(4*n)    #theta * X[i] - t[i], -t[i], z1[i], z2[i]
    z1, z2 = nvar, nvar+n
    zcon = ncon
    thetaafe, tafe, z1afe, z2afe = nafe, nafe+n, nafe+2*n, nafe+3*n

    # z1 + z2 = 1
    task.putaijlist(range(zcon, zcon+n), range(z1, z1+n), [1]*n)
    task.putaijlist(range(zcon, zcon+n), range(z2, z2+n), [1]*n)
    task.putconboundsliceconst(zcon, zcon+n, boundkey.fx, 1, 1)
    task.putvarboundsliceconst(nvar, nvar+2*n, boundkey.fr, -inf, inf)

    # Affine conic expressions
    afeidx, varidx, fval = [], [], []

    ## Thetas
    for i in range(n):
      for j in range(d):
        afeidx.append(thetaafe + i)
        varidx.append(theta + j)
        fval.append(-X[i][j] if y[i]==1 else X[i][j])

    # -t[i]
    afeidx.extend([thetaafe + i for i in range(n)] + [tafe + i for i in range(n)])
    varidx.extend([t + i for i in range(n)] + [t + i for i in range(n)])
    fval.extend([-1.0]*(2*n))

    # z1, z2
    afeidx.extend([z1afe + i for i in range(n)] + [z2afe + i for i in range(n)])
    varidx.extend([z1 + i for i in range(n)] + [z2 + i for i in range(n)])
    fval.extend([1.0]*(2*n))

    # Add the expressions
    task.putafefentrylist(afeidx, varidx, fval)

    # Add a single row with the constant expression "1.0"
    oneafe = task.getnumafe()
    task.appendafes(1)
    task.putafeg(oneafe, 1.0)

    # Add an exponential cone domain
    expdomain = task.appendprimalexpconedomain()
    
    # Conic constraints
    for i in range(n):
      task.appendacc(expdomain, [z1afe+i, oneafe, thetaafe+i], None)
      task.appendacc(expdomain, [z2afe+i, oneafe, tafe+i], None)

Once we have this subroutine, it is easy to implement a function that builds the regularized loss function model (11.13).

Listing 11.8 Implementation of (11.13). 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
def logisticRegression(env, X, y, lamb=1.0):
    n, d = int(X.shape[0]), int(X.shape[1])         # num samples, dimension
    
    with env.Task() as task:
        # Variables [r; theta; t; u]
        nvar = 1+d+2*n
        task.appendvars(nvar)
        task.putvarboundsliceconst(0, nvar, boundkey.fr, -inf, inf)
        r, theta, t = 0, 1, 1+d

        # Objective lambda*r + sum(t)
        task.putcj(r, lamb)
        task.putclist(range(t, t+n), [1.0]*n)

        # Softplus function constraints
        softplus(task, d, n, theta, t, X, y);

        # Regularization
        # Append a sequence of linear expressions (r, theta) to F
        numafe = task.getnumafe()
        task.appendafes(1+d)
        task.putafefentry(numafe, r, 1.0)
        for i in range(d):
          task.putafefentry(numafe + i + 1, theta + i, 1.0)
        
        # Add the constraint
        task.appendaccseq(task.appendquadraticconedomain(1+d), numafe, None)

        # Solution
        task.optimize()
        xx = task.getxxslice(soltype.itr, theta, theta+d)

        return xx

Example: 2D dataset fitting

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.

_images/logistic-regression.png

Fig. 11.2 Logistic regression example with none, medium and strong regularization (small, medium, large \(\lambda\)). Without regularization we get obvious overfitting.