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 11.8 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
MSKrescodee softplus(MSKtask_t task, int d, int n, MSKint32t theta, MSKint32t t, double* X, int* y)
{
  MSKint32t nvar, ncon;
  MSKint64t nafe, thetaafe, tafe, z1afe, z2afe, oneafe, expdomain;
  MSKint32t z1, z2, zcon, v1con, v2con;
  MSKint32t  *subi = (MSKint32t*) calloc(2*n, sizeof(MSKint32t));
  MSKint32t  *subj = (MSKint32t*) calloc(3*n, sizeof(MSKint32t));
  MSKrealt   *aval = (MSKrealt*) calloc(2*n, sizeof(MSKrealt));
  MSKint64t  *afeidx = (MSKint64t*) calloc(d*n+4*n, sizeof(MSKint64t));
  MSKint32t  *varidx = (MSKint32t*) calloc(d*n+4*n, sizeof(MSKint32t));
  MSKrealt   *fval   = (MSKrealt*) calloc(d*n+4*n, sizeof(MSKrealt));
  MSKint64t  idx[3];
  int        k, i, j;
  MSKrescodee res = MSK_RES_OK;

  MSKCALL(MSK_getnumvar(task, &nvar));
  MSKCALL(MSK_getnumcon(task, &ncon));
  MSKCALL(MSK_getnumafe(task, &nafe));
  MSKCALL(MSK_appendvars(task, 2*n));   // z1, z2
  MSKCALL(MSK_appendcons(task, n));     // z1 + z2 = 1
  MSKCALL(MSK_appendafes(task, 4*n));   //theta * X[i] - t[i], -t[i], z1[i], z2[i]

  z1 = nvar, z2 = nvar+n;
  zcon = ncon;
  thetaafe = nafe, tafe = nafe+n, z1afe = nafe+2*n, z2afe = nafe+3*n;
  
  // Linear constraints
  k = 0;
  for(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++;
  }
  MSKCALL(MSK_putaijlist(task, 2*n, subi, subj, aval));
  MSKCALL(MSK_putconboundsliceconst(task, zcon, zcon+n, MSK_BK_FX, 1, 1));
  MSKCALL(MSK_putvarboundsliceconst(task, nvar, nvar+2*n, MSK_BK_FR, -inf, inf));

  // Affine conic expressions
  k = 0;

  // Thetas
  for(i = 0; i < n; i++) {
    for(j = 0; j < d; j++) {
      afeidx[k] = thetaafe + i; varidx[k] = theta + j; 
      fval[k] = ((y[i]) ? -1 : 1) * X[i*d+j];
      k++;
    }
  }

  // -t[i]
  for(i = 0; i < n; i++) {
    afeidx[k] = thetaafe + i; varidx[k] = t + i; fval[k] = -1; k++;
    afeidx[k] = tafe + i;     varidx[k] = t + i; fval[k] = -1; k++;
  }

  // z1, z2
  for(i = 0; i < n; i++) {
    afeidx[k] = z1afe + i; varidx[k] = z1 + i; fval[k] = 1; k++;
    afeidx[k] = z2afe + i; varidx[k] = z2 + i; fval[k] = 1; k++;
  }

  // Add the expressions
  MSKCALL(MSK_putafefentrylist(task, d*n+4*n, afeidx, varidx, fval));

  // Add a single row with the constant expression "1.0"
  MSKCALL(MSK_getnumafe(task, &oneafe));
  MSKCALL(MSK_appendafes(task,1));
  MSKCALL(MSK_putafeg(task, oneafe, 1.0));

  // Add an exponential cone domain
  MSKCALL(MSK_appendprimalexpconedomain(task, &expdomain));

  // Conic constraints
  for(i = 0; i < n; i++)
  {
    idx[0] = z1afe+i, idx[1] = oneafe, idx[2] = thetaafe+i;
    MSKCALL(MSK_appendacc(task, expdomain, 3, idx, NULL));
    idx[0] = z2afe+i, idx[1] = oneafe, idx[2] = tafe+i;
    MSKCALL(MSK_appendacc(task, expdomain, 3, idx, NULL));
  }
    
  free(subi); free(subj); free(aval); 
  free(afeidx); free(varidx); free(fval); 
  return res;
}

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

Listing 11.9 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
MSKrescodee logisticRegression(MSKenv_t       env,
                               int            n,    // num samples
                               int            d,    // dimension
                               double        *X, 
                               int           *y,
                               double         lamb,
                               double        *thetaVal)   // result
{
  MSKrescodee res = MSK_RES_OK;
  MSKrescodee trm = MSK_RES_OK;
  MSKtask_t task = NULL;
  MSKint32t nvar = 1+d+n;
  MSKint32t r = 0, theta = 1, t = 1+d;
  MSKint64t numafe, quadDom;
  int i = 0;

  MSKCALL(MSK_maketask(env, 0, 0, &task));
  MSKCALL(MSK_linkfunctotaskstream(task, MSK_STREAM_LOG, NULL, printstr));

  // Variables [r; theta; t]
  MSKCALL(MSK_appendvars(task, nvar));
  MSKCALL(MSK_putvarboundsliceconst(task, 0, nvar, MSK_BK_FR, -inf, inf));

  // Objective lambda*r + sum(t)
  MSKCALL(MSK_putobjsense(task, MSK_OBJECTIVE_SENSE_MINIMIZE));
  MSKCALL(MSK_putcj(task, r, lamb));
  for(i = 0; i < n && res == MSK_RES_OK; i++) 
    MSKCALL(MSK_putcj(task, t+i, 1.0));

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

  // Regularization
  // Append a sequence of linear expressions (r, theta) to F
  MSKCALL(MSK_getnumafe(task, &numafe));
  MSKCALL(MSK_appendafes(task,1+d));
  MSKCALL(MSK_putafefentry(task, numafe, r, 1.0));
  for(i = 0; i < d; i++)
    MSKCALL(MSK_putafefentry(task, numafe + i + 1, theta + i, 1.0));

  // Add the constraint
  MSKCALL(MSK_appendquadraticconedomain(task, 1+d, &quadDom));
  MSKCALL(MSK_appendaccseq(task, quadDom, 1+d, numafe, NULL));

  // Solution
  MSKCALL(MSK_optimizetrm(task, &trm));
  MSKCALL(MSK_solutionsummary(task, MSK_STREAM_MSG));

  MSKCALL(MSK_getxxslice(task, MSK_SOL_ITR, theta, theta+d, thetaVal));
  
  return res;
}

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.

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