Linear Regression Demo and Derivation

This page explains how gradient descent works by using it to compute a line that best goes through some given data. This line will be referred to as the "line of best fit."

Input data

Click below to add points that make roughly a line. Then click "compute line of best fit" to see the line of best fit computed using the selected parameters.



How it works

The goal is to find the line that's closest to the given points. A line, \(\hat{y} = mx + b\) is determined by it's slope (m) and it's intercept (b). To determine which line is closest, we can define an error function that will tell us how far off we are and then minimize it to get the closest line. One way we could define such an error function is to take the sum of the squares of the distances of each of the points: \[E(m, b) = \sum (y - \hat{y})^2 = \sum_{(x,y) \in data} (y - mx - b)^2\] (One reason to square the difference is to make sure the values are all positive. Additionally, it's easier to work with a squared quantity rather than its absolute value.) Notice this error is a function of \(m\), \(b\), and the data. So given some data, an \(m\), and a \(b\) we can determine how close the line \(\hat{y} = mx + b\) is to the data. If the line goes through all the points exactly, the distance for each will be zero and so the error will be zero. If the line is way off from the points, the distances will be large and the error will be large. So our task of finding the closest line is a matter of finding the \(m\) and \(b\) that minimize the error.

Minimizing the error

There are few ways to approach this problem and a few ways to find the closest line. We could take a statistical approach or even a linear algebra approach (SVD) to find the best line. However, we're going to take an iterative approach to approximate the solution. Why? It's true that we could use some fancy methods to solve this "line of best fit" problem, but what if we wanted to find the best quadratic function through the data, or perhaps something much more complicated like an artificial neural network? In these cases finding an exact solution may be hard, if not impossible (that is, not having a closed form solution) and require an iterative numerical solution. The method we'll be using is called Gradient Descent and usually does a good job of minimizing error functions (though it is not guaranteed to give the most minimal solution, more about this later).

Gradient Descent: Climbing down the mountain

Gradient descent is a popular technique for minimizing general functions in an iterative manner. Let's consider the error function again for some fixed data: it is a function of two variables, \(m\) and \(b\), that outputs a number, which indicates how close or far the line is from the data. Bigger numbers mean the line is further away.

Since this error function is a function of two variables, we can imagine it like a mountain range: there are some coordinates \(m\) and \(b\) (you can think like latitude and longitude) and then the result is the altitude. The "line of best fit" is the place on the map (the \(m\) and \(b\) coordinates) that has the lowest altitude. In real life, if we were on a mountain, we could probably look and see where the point of lowest altitude is located. However, each glance at the error function requires computation, so looking around for the lowest altitude will require lots of computation and would be a slow and an inefficient method.

What we can do instead is to get a feel for how the altitude is changing in our local area (is our left foot down and right foot up? Do we feel like we're leaning in some direction?). We can do this with the error function as well by computing something called the gradient. The gradient of a scalar function (a function that outputs a single number) at a particular point is a vector that points in the direction of greatest increase. In our mountain range example, the gradient would tell us, given where we are, which direction would be the steepest (and also some information about how steep). So, if we wanted to find our way down the mountain, we should go in the opposite direction of the gradient.

It is important that we go down the mountain in small steps. This is because the direction down the mountain might twist and turn as we walk. Small steps are important because taking too big of a step might cause us to miss a quick way down the mountain. This process of step-by-step walking in the opposite direction of the gradient is called gradient descent.

Local vs Global Minima and Variations

Gradient descent is pretty straightforward: to get off the mountain, follow the path that slopes down the most. Beware, however, this doesn't necessarily get you to ground level; you could get stuck in a local minima. Imagine in this mountain range there is a valley between two mountains, if you end up there, there's no longer a path downhill and the gradient descent approach will be stuck there. A valley like this is called a local minima because it is the minimum value locally (compared to the nearby points). This is in contrast to the global minima which is actually the minimum value. Finding the global minima is a bit trickier, though it is not always necessary. Sometimes the results of a local minima are fine, and, if not, doing gradient descent again from a different starting location might give a better minima. There are variations to gradient descent that attempt to address the local minima problem and other variations to make things less computationally demanding. A popular variant is stochastic gradient descent which avoids calculating the complete gradient by randomly selecting a few of the components and using those. In the imaginary mountain case this would equate to randomly selecting either the N-S direction or the E-W direction, then taking the appropriate downhill step in whichever direction. In problems with more variables this random subselection could be a big computational time saver.

How to Compute the Gradient

This section is particularly math intensive and requires knowledge of calculus (probably just one semester of calculus is sufficient). Most programmers probably can skip this section because a lot of software (like TensorFlow for example) uses automatic differentiation to automatically compute the gradient (this is one of the reasons TensorFlow has a special "Variable" type so it can automatically do the math). That being said, there isn't too much math here, you just need to know about derivatives, which basically describe how functions change given changes in their variables.

The gradient of some \(f(x, y)\) is defined as \[\nabla f(x,y) = \begin{bmatrix} \frac{\partial f}{\partial x}(x,y) \\ \frac{\partial f}{\partial y}(x,y) \end{bmatrix}\] (If you haven't seen those \(\partial\) symbols before, it's called partial differentiation and it just means treat the other variables as constants when you take the derivative.)

So in the case of "line of best fit" we need to compute the gradient of our error function. \[\nabla E = \begin{bmatrix} \frac{\partial E}{\partial m} \\ \frac{\partial E}{\partial b} \end{bmatrix}\] \[\frac{\partial E}{\partial m} = \sum_{(x,y) \in data} -2(y - mx - b)x \] \[\frac{\partial E}{\partial b} = \sum_{(x,y) \in data} -2(y - mx - b) \] Then to find the "line of best fit" we can start with some random initial guess \(\begin{bmatrix} m_0 \\ b_0 \end{bmatrix}\) and iterate several times over \(\begin{bmatrix} m_{n+1} \\ b_{n+1} \end{bmatrix} = \begin{bmatrix} m_n \\ b_n \end{bmatrix} + \alpha \begin{bmatrix} \frac{\partial E}{\partial m} \\ \frac{\partial E}{\partial b} \end{bmatrix}\), where \(\alpha\) is our "mountain step" size, called the learning rate, which may need to be tweaked (but is usually something like \(0.03\)). Depending on the problem, the number of iterations needed may be in the hundreds, thousands, or even more as is the case for artificial neural networks.