Logistic regression is a classification case of linear regression whith dependent variable $y$ taking binary values.

Problem: Given a training set $\langle x^{(i)}, y^{(i)} \rangle$, $1 \le i \le m$, $x \in \mathbb{R}^{n+1}$, $x^{(i)} _ 0 = 0$, $y^{(i)} \in $ {0,1}, find classification function

\[h_\theta(x) = P(y = 1 | x; \theta)\]

Gradient Descent

Let’s build function $h_\theta(x)$ as a sigmoid function of $\theta\cdot x$

\[h_\theta(x) = g(\theta\cdot x), \quad g(z) = \mathbb{sigmoid}(z) = \frac{1}{1 + e^{-z}}\]

Sigmoid function has rank infinity, i.e. it operates on scalars, vectors and matrices.

function g = sigmoid(z)
    g = 1 ./ (1 + exp(-z));
end

To find optimal parameter $\theta \in \mathbb{R}^{n+1}$ we are going to use optimized gradient descent method which takes as arguments cost function $J(\theta)$ and its gradient. For logistic regression they are

\[J(\theta) = -\frac{1}{m} \left( y^T \ln h_\theta(X) + (1-y)^T \ln (1-h_\theta(X)) \right) \\ \nabla J(\theta) = \frac{1}{m} X^T (h_\theta(X) - y)\]

where $X = (x^{(i)}_j) _{m \times n+1}$ is a matrix of the training examples from the previous lecture.

Analogous to linear regression, logistic regression can be regularized too

\[J(\theta) = -\frac{1}{m} \left( y^T \ln h_\theta(X) + (1-y)^T \ln (1-h_\theta(X)) \right) + \frac{\lambda}{2m} \| E\theta \|^2 \\ \nabla J(\theta) = \frac{1}{m} \left( X^T (h_\theta(X) - y) + \lambda E\theta \right)\] 
function [J, grad] = costFunction(theta, X, y, lambda)
    m = length(y); % number of training examples
    h = sigmoid(X * theta);
    J = (y' * log(h) + (1 - y)' * log(1 - h)) / -m;
    grad = X' * (h - y) / m;

    % Regularization
    th = theta; th(1) = 0;

    J = J + th' * th * lambda / m / 2;
    grad = grad + th * lambda / m;
end

Having computed $\theta$ we can now implement the prediction function

function p = predict(theta, X)
    p = sigmoid(X * theta) >= 0.5;
end

which can be used to classify new examples and check the prediction accuracy on the training set

function a = accuracy(theta, X, y)
    p = predict(theta, X);
    a = mean(double(p == y)) * 100;
end

Multi-class Classification

Logistic regression works for binary $y$. Suppose now that $y^{(i)} \in ${$1,…,K$}, where $K > 2$. In this case we can use One-vs-All variation of this algorithm.

Step 1. Convert vector $y$ into a binary matrix $Y$

\[y = \begin{pmatrix} y^{(1)} \\ y^{(2)} \\ y^{(3)} \\ \vdots \\ y^{(m)} \\ \end{pmatrix} \quad \rightarrow \quad Y = \begin{pmatrix} y^{(1)}_1 & y^{(1)}_2 & \cdots & y^{(1)}_K \\ y^{(2)}_1 & y^{(2)}_2 & \cdots & y^{(2)}_K \\ y^{(3)}_1 & y^{(3)}_2 & \cdots & y^{(3)}_K \\ \vdots & \vdots & \ddots & \vdots \\ y^{(m)}_1 & y^{(m)}_2 & \cdots & y^{(m)}_K \\ \end{pmatrix}\]

where $y^{(i)}_k = \delta _{k y^{(i)}}$, i.e. $y^{(i)}_k = 1$ when $y^{(i)} = k$, otherwise $y^{(i)}_k = 0$.

Step 2. Train logistic classifier on every column of matrix $Y$. The result will be a matrix $\Theta = (\theta_{jk})_{n+1 \times K}$

\[\Theta = \begin{pmatrix} \theta_{01} & \theta_{02} & \cdots & \theta_{0K} \\ \theta_{11} & \theta_{12} & \cdots & \theta_{1K} \\ \theta_{21} & \theta_{22} & \cdots & \theta_{2K} \\ \vdots & \vdots & \ddots & \vdots \\ \theta_{n1} & \theta_{n2} & \cdots & \theta_{nK} \\ \end{pmatrix}\]

Step 3. For any given vector $x$ compute vector $h = x^T \Theta$. Then the predicted value $y$ will be

\[y = \{ p \: | \: h_p = \mathbb{max} (h_k), 1 \le k \le K \}\]

To compute accuracy of the one-vs-all classifier on the training set use accuracy.m script from above with modified predict.m

function p = predict(Theta, X)
    a = sigmoid(X * Theta);
    [v, p] = max(a, [], 2);
end