This is one of the most popular optimisation algorithms in Data Science. Do you know how it works?
Why should I read this?
Most of Machine Learning models use some sort of optimisation algorithm to find the parameters that yield the smallest error possible, and Gradient Descent is probably one of the most popular of those algorithms.
If you are a Data Scientist or work with Machine Learning, you certainly work with some algorithm that uses Gradient Descent. Even though you probably will never have to hard code it yourself, understanding how it works can help you understand why your model is not giving good results, or even not converging at all.
How does it work?
Gradient Descent finds the parameters that result in the local minimum of a function. In ML algorithms, this function is usually some sort of cost function, related to the difference between the model output and the actual observations.
Gradient Descent chooses a random point in the function and starts “walking” towards the lowest point around, until it reaches a local minimum. The starting point is a random combination of parameters, that yield a certain cost (error). The following step is to find a point nearby where this cost is lower, and move to it.
Imagine you are walking on a mountain in the dark, trying to find the lowest point possible. You slowly feel the way, trying to always walk down. The more gentle the slope gets, the slower you walk, since the more you expect it to be reaching an end.
Making it formal
How do you do control that speed when searching in a function, however? By using the derivative of the function on that specific point.
You will then sequentially update your function parameters θ with the derivative of the cost function J(Θ) on the point you are right now, adjusted by ⍺.
⍺ is the learning rate (how much you adjust the size of the “step” when the algorithm “walks”). If ⍺ is too small, the algorithm can be too slow. If ⍺ is is too big, the algorithm can fail to converge or even diverge.
:= is an assignment operator, meaning we assign such value to our variable.
So, ideally, you want your learning rate to be as high as possible (so your algorithm runs as fast as possible), without being so high that it diverges. Next time your model doesn’t converge, try using a smaller learning rate.
For really big datasets, Gradient Descent can be really computationally expensive, since it has to calculate J through the whole dataset for every iteration. So, let’s look at some alternatives that work better for a lot of data.
Stochastic Gradient Descent
Stochastic Gradient Descent starts by randomly shuffling your data. This is important because it will read through the observations step by step, so they shouldn’t be in any particular order. Then, it will calculate the cost function only for the first observation and take the first step towards the minimum based only on that observation. For the second iteration, it will do the same, based solely on the second observation. And so on, until it reaches the last observation in the training set. If you have a lot of data, this should still yield satisfactory results within reasonable time.
Mini-Batch Gradient Descent
The Mini-Batch version is somewhere in between the Stochastic and the classic ones. Here, instead of using only one training example at each iteration, it uses a mini-batch of size b, a user-defined parameter. This means that if we set b = 10, for example, we would be doing a Gradient Descent using 10 observations at each iteration. This can sometimes actually be faster than the Stochastic version. If you think about it, you will see that the Stochastic version is actually a special case of the Mini-Batch, when b = 1.
Now that you know how Gradient Descent works, you know how to fix it when it doesn’t converge, by changing the learning rate, and how to make it faster, by implementing its alternative versions.