Why Logistic Regression Beats Linear Regression for Classification

In machine learning, there are two main types of tasks: regression and classification. Linear Regression is designed for regression tasks where the goal is to predict continuous values (like predicting house prices). However, people sometimes try to use it for classification tasks, which aim to assign discrete labels to data (e.g., classifying if an email is spam or not). While it might seem like a simple solution, using Linear Regression for classification is often a bad idea, and a more suitable method is Logistic Regression.

Why Linear Regression Fails in Classification

1. Linear Regression Outputs Continuous Values, Not Probabilities

Linear Regression predicts continuous values, meaning the result could be any real number. For example, if you are trying to classify emails as “spam” or “not spam,” you want the output to be either 0 (not spam) or 1 (spam). But with Linear Regression, you might get values like 3.567, -1.24, or 0.43. These values are continuous and don’t naturally translate into discrete class labels like 0 or 1.

Example:

• Suppose Linear Regression predicts the following values: [1.23, 0.43, 4.32, 3.49].
• If we want to classify these into six categories (0 to 5), we can round them: [1, 0, 4, 3].
• While this is a hacky way to use Linear Regression for classification, it’s imprecise and doesn’t work well with all datasets.

2. Linear Regression is Sensitive to?Outliers

Linear Regression tries to fit a straight line through your data. However, this line can be easily skewed by outliers?—?data points that don’t follow the general pattern. These outliers can pull the decision boundary in the wrong direction, making the model perform poorly on new, unseen data.

Example with Tumor Data: Let’s say you’re using Linear Regression to classify tumors as malignant (cancerous) or benign (non-cancerous) based on size. Initially, your model might look like this:

In this case, it seems like the model is working well: if the size is small, the tumor is benign (0), and if it’s large, it’s malignant (1). But what if you add a new data point where a very large tumor is benign? Now, the line shifts, making incorrect predictions for other data points:

This shift happens because Linear Regression tries to fit the best line for all data points, and outliers heavily influence that line.

3. No Decision?Boundary

In classification tasks, it’s important to have a decision boundary, which is the threshold that separates different classes. Linear Regression doesn’t create a decision boundary as effectively as Logistic Regression because it predicts continuous values without clear thresholds for classification.

Example: Imagine you’re classifying tumors as benign (0) or malignant (1). With Logistic Regression, the algorithm creates a clear decision boundary, represented by a line that separates the two classes:

Here, tumors on one side of the line are predicted to be malignant, and those on the other side are predicted to be benign. This clean separation is hard to achieve with Linear Regression.

Why Logistic Regression is Better for Classification

Logistic Regression is specifically designed for classification tasks. It solves the issues that Linear Regression faces when used for classification.

1. Logistic Regression Outputs Probabilities

Instead of predicting continuous values, Logistic Regression predicts probabilities. It uses a function called the sigmoid function, which takes any input and converts it into a probability between 0 and 1.

This makes Logistic Regression ideal for binary classification. The output tells you how likely it is that a data point belongs to the “positive” class. For example, if the sigmoid function outputs 0.8, there’s an 80% chance the point is in the positive class.

2. Decision Boundary for Classification

Logistic Regression establishes a clear decision boundary. The decision boundary separates the classes based on the probabilities calculated by the sigmoid function. In binary classification, any value greater than 0.5 is classified as “1” (positive class), and anything less than 0.5 is classified as “0” (negative class).

Example: Returning to the tumor classification example, Logistic Regression would calculate the probability of a tumor being malignant based on its size. The decision boundary (where the probability equals 0.5) clearly separates benign and malignant tumors:

3. Resistant to?Outliers

Logistic Regression is less affected by outliers than Linear Regression. While it still considers all data points, its focus is on correctly classifying points based on their distance from the decision boundary. This makes it a better fit for classification tasks where data may contain outliers or noise.

Summary?: Linear Regression is great for regression tasks but struggles with classification due to continuous outputs and sensitivity to outliers. Logistic Regression, on the other hand, is designed for classification problems. It produces clear probabilities, creates a solid decision boundary, and is robust to outliers, making it a far more suitable algorithm for binary (and even multi-class) classification problems.