Gradient Descent for Logistic Regression

Unlike linear regression, logistic regression does not have a closed-form solution. Instead, we use the generalized linear model approach using gradient descent and maximum likelihood.

First, lets discuss logistic regression. Unlike linear regression, values in logistic regression generally take two forms, log-odds and probability. Log-odds is the value returned when we multiply each term by its coefficient and sum the results. This value can span from -Inf to Inf.

Probability form takes the log-odds form and squishes it to values between 0 and 1. This is important because logistic regression is a binary classification method which returns the probability of an event occurring.

To transform log-odds to a probability we perform the following operation: exp(log-odds) / 1 + exp(log-odds). And to transform probability back to log odds we perform the following operation: log(probability / 1 – probability).

Next, we need to consider our cost function. All generalized linear models have a cost function. For logistic regression, we maximize likelihood. To compute the likelihood of a set of coefficients we perform the following operations: sum(log(probability)) for data points with a true classification of 1 and sum(log(1 – probability)) for data points with a true classification of 0.

Even though we can compute the given cost of a set of parameters, how can we determine which direction will improve our outcome? It turns out we can take the partial derivative for each parameter (b0, b1, … bn) and nudge our parameters into the right direction.

Suppose we have a simple logistic regression model with only two parameters, b0 (the intercept) and b1 (the relationship between x and y). We would compute the gradient of our parameters using the following operations: b0 – rate * sum(probability – class) for the intercept and b1 – rate * sum((probability – class) * x)) for the relationship between x and y.

Note that rate above is the learning rate. A larger learning rate will nudge the coefficients more quickly where a smaller learning rate will approach the coefficients more slowly, but may achieve better estimates.

Now lets put all of this together! The Python function to perform gradient descent for logistic regression is surprisingly simple and requires the use of only Numpy. We can see gradient descent in action in the visual below which shows the predicted probabilities for each iteration.

import numpy as np

def descend(x, y, b0, b1, rate):

    # Determine x-betas
    e_xbeta = np.exp(b0 + b1 * x)
    x_probs = e_xbeta / (1 + e_xbeta)
    p_diffs = x_probs - y

    # Find gradient using partial derivative
    b0 = b0 - (rate * sum(p_diffs))
    b1 = b1 - (rate * sum(p_diffs * x))
    return b0, b1

def learn(x, y, rate=0.001, epoch=1e4):

    # Initial conditions
    b0 = 0 # Starting b0
    b1 = 0 # Starting b1
    epoch = int(epoch)

    # Arrays for coefficient history
    b0_hist = np.zeros(epoch)
    b1_hist = np.zeros(epoch)

    # Iterate over epochs
    for i in range(epoch):
        b0, b1 = descend(x, y, b0, b1, rate)
        b0_hist[i] = b0
        b1_hist[i] = b1

    # Returns history of parameters
    return b0_hist, b1_hist

# Data for train
x = np.array([0, 1, 2, 3, 4, 3, 4, 5, 6, 7])
y = np.array([0, 0, 0, 0, 0, 1, 1, 1, 1, 1])

# Generate model
b0_hist, b1_hist = learn(x, y)

Implementing KNN in Python

K-nearest neighbors (KNN) is an algorithm which identifies the k nearest data points in a training sample to a new observation. Typically, nearest is defined by the Euclidian (or straight line) distance, however, other distance norms can be used.

Python is already home to several KNN implementations the most famous of which is the scikit-learn implementation. I still believe there is value in writing your own model implementations to learn more about how they work.

First lets break down what KNN is doing visually and then code up our own implementation. The visual below (built using D3.js) shows several points which are classified into the red and blue groups.

You can hover your mouse over this visual to develop an understanding of how the nearest three points impact the classification of the point.

We can identify the three (k = 3) closest points and determine of those, which classification is the most common. The most common classification becomes our predicted value.

A few notes before we jump into our own implementation. First, it is common to use an odd number for k when performing classification to avoid ties. Second, one downside of KNN when compared to other models is that KNN must be packaged with the training data to make predictions. This is different than linear regression which only requires the coefficients to be known at the time of prediction, for example.

Now let’s look at my implementation of KNN in Python. Only 8 lines of code (excluding function imports)! A safer version of this code may also include several assertion checks to ensure inputs are of the expected type and shape.

import numpy as np
import scipy as sci

def knn(new, train, labels, k=3, mode="c"):

    distances = np.sum((new - train) ** 2, axis=1)
    k_closest = distances.argsort()[:k]
    values = np.take(labels, k_closest)
    if mode == "c":
        return sci.stats.mode(values)[0][0]
    if mode == "r":
        return np.mean(values)

Lets look at this function line by line. First, I define a function called knn which accepts a singular new observation called new, the training data called train with its associated labels (the correct prediction), and the mode which is either c for classification or r for regression.

def knn(new, train, labels, k=3, mode="c")

From there I compute how far each of the training points is from the new observation. To accurately compute the distances you would need to take the square root of this value. However, because we are only interested in the rank ordering of points, we can skip that step.

distances = np.sum((new - train) ** 2, axis=1)

Next I use argsort and take from numpy to rank order the indices by how close they are to the new observation. I use index slicing to grab the k nearest points. From there I use take to grab the values of the k closest indices from the label data.

k_closest = distances.argsort()[:k]
values = np.take(labels, k_closest)

Finally, I take the mode of the values for classification or the mean for regression. To predict over multiple observations I could pass the function into a list comprehension:

[knn(i, train, labels) for i in test]

This was a simple overview of KNN regression using basic numpy and scipy functions!