How Linear Regression Learns

Okay, we've established that linear regression aims to find a straight line that best represents the relationship between an input feature (let's call it $x$) and a continuous output value ($y$). The equation for this line is typically written as:

$y = mx + b$

Here, $m$ represents the slope of the line (how much $y$ changes for a one-unit increase in $x$), and $b$ is the y-intercept (the value of $y$ when $x$ is zero). But how does the algorithm actually find the specific values for $m$ and $b$ that result in the "best" line for our data? This is the "learning" part.

Think of it like tuning an instrument. You adjust the tuning pegs (the parameters $m$ and $b$) until the sound (the fit to the data) is just right.

The Goal: Minimizing Error

The "best" line isn't subjective; in machine learning, "best" usually means the line that minimizes the overall error between its predictions and the actual data points. Imagine you have your data points plotted on a graph. For any given line $y = mx + b$, you can calculate:

1. The Predicted Value: For each actual data point $(x_i, y_i)$, plug the $x_i$ value into the line's equation to get a predicted value, let's call it $\hat{y}_i = mx_i + b$.
2. The Error (Residual): Calculate the difference between the actual value $y_i$ and the predicted value $\hat{y}_i$. This difference, $y_i - \hat{y}_i$, is the error or residual for that specific point. It's essentially the vertical distance between the data point and the line on the graph.

The linear regression algorithm's goal is to find the values of $m$ and $b$ that make the total error across all data points as small as possible.

This plot shows sample data points, an initial guess for the line, and a line that fits the data much better after the learning process. The goal is to find the line like the green one.

The Learning Mechanism: Iterative Adjustment

Computers aren't great at just "seeing" the best line like humans sometimes can. Instead, algorithms like linear regression use an iterative process:

1. Initialization: Start with some initial guess for the parameters $m$ and $b$. These could be random values, or simply zero ($m=0, b=0$). This gives us an initial line.
2. Error Calculation: Calculate the total error for the current line using all the data points. We need a precise way to measure this total error, which is where a cost function comes in (we'll look at this in detail in the next section). For now, just know it gives us a single number representing how "bad" the current line's fit is.
3. Parameter Update: Based on the calculated error, the algorithm intelligently adjusts the values of $m$ and $b$. The key is to adjust them in a direction that decreases the total error. Think of it as nudging the slope and intercept slightly to make the line fit better.
4. Repeat: Go back to step 2 with the new values of $m$ and $b$. Calculate the new, hopefully smaller, error. Update the parameters again. Repeat this process many times.

With each iteration, the line should get progressively closer to the optimal fit, and the total error should decrease. This continues until the error is small enough, or until the changes in $m$ and $b$ become negligible, meaning the algorithm has converged on the best values it can find.

The specific mathematical technique often used to figure out how to adjust $m$ and $b$ in step 3 is called Gradient Descent, which we will explore later in this chapter. For now, the important concept is this iterative process of measuring error and adjusting parameters to minimize that error. That's how linear regression "learns" the best line from the data you provide.