When I first started out learning about machine learning algorithms, it turned out to be quite a task to gain an intuition of what the algorithms are doing. Not just because it was difficult to understand all the mathematical theory and notations, but it was also plain boring. When I turned to online tutorials for answers, I could again only see equations or high level explanations without going through the detail in a majority of the cases.
It was then that one of my data science colleagues introduced me to the concept of working out an algorithm in an excel sheet. And that worked wonders for me. Any new algorithm, I try to learn it in an excel at a small scale and believe me, it does wonders to enhance your understanding and helps you fully appreciate the beauty of the algorithm.
Let me explain the above using an example.
Most of the data science algorithms are optimization problems and one of the most used algorithms to do the same is the Gradient Descent Algorithm.
Now, for a starter, the name itself Gradient Descent Algorithm may sound intimidating, well, hopefully after going though this post, that might change.
Lets take the example of predicting the price of a new price from housing data.
Given historical housing data, the task is to create a model that predicts the price of a new house given the house size.
The task — for a new house, given its size (X), what will its price (Y) be?
Lets start off by plotting the historical housing data:
Now, we will use a simple linear model, where we fit a line on the historical data, to predict the price of a new house (Ypred) given its size (X)
In the above chart, the red line gives the predicted house price(Ypred) given house size(X). Ypred = a+bX
The blue line gives the actual house prices from historical data (Yactual)
The difference between Yactual and Ypred (given by the yellow dashed lines) is the prediction error (E)
So, we need to find a line with optimal values of a,b (called weights) that best fits the historical data by reducing the prediction error.
So, our objective is to find optimal a, b that minimizes the error between actual and predicted values of house price:
Sum of Squared Errors (SSE) = ½ (Actual House Price — Predicted House Price)²
= ½(Y — Ypred)²
(Please note that there are other measures of Error. SSE is just one of them.)
This is where Gradient Descent comes into the picture. Gradient descent is an optimization algorithm that finds the optimal weights (a,b) that reduces prediction error.
Lets now go step by step to understand the Gradient Descent algorithm:
Step 1: Initialize the weights (a & b) with random values and calculate Error (SSE)
Step 2: Calculate the gradient i.e. change in SSE when the weights (a & b) are changed by a very small value from their original randomly initialized value. This helps us move the values of a & b in the direction in which SSE is minimized.
Step 3: Adjust the weights with the gradients to move towards the optimal values where SSE is minimized
Step 4: Use the new weights for prediction and to calculate the new SSE
Step 5: Repeat steps 2 and 3 till further adjustments to weights doesn’t significantly reduce the Error
We will now go through each of the steps in detail (I have done the above steps in excel, which I have pasted below). But before that, we have to standardize the data as it makes the optimization process faster.
Step 1: To fit a line Ypred = a + b X, start off with random values of a and b and calculate prediction error (SSE)
Step 2: Calculate the error gradient w.r.t the weights
∂SSE/∂a =-(Y-YP)
∂SSE/∂b =-(Y-YP)X
Here, SSE=½ (Y-YP)² = ½(Y-(a+bX))²
A bit of calculus here, but that’s about it!!
∂SSE/∂a and ∂SSE/∂b are the gradients and they give the direction of the movement of a,b w.r.t to SSE.
Step 3: Adjust the weights with the gradients to reach the optimal values where SSE is minimized
We need to update the random values of a,b so that we move in the direction of optimal a, b.
Update rules:
1) a -∂SSE/∂a
2) b -∂SSE/∂b
So, update rules:
here, r is the learning rate = 0.01, which is the pace of adjustment to the weights.
Step 4: Use new a and b for prediction and to calculate new Total SSE
You can see with the new prediction, the total SSE has gone down (0.677 to 0.553). That means prediction accuracy has improved.
Step 5: Repeat step 3 and 4 till the time further adjustments to a, b doesn’t significantly reduces the error. At that time, we have arrived at the optimal a,b with the highest prediction accuracy.
This is the Gradient Descent Algorithm. This optimization algorithm and its variants form the core of many machine learning algorithms like Neural Networks and even Deep Learning.
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Comment
I worked on neural networks (a.k.a. "Deep Learning"), pre "McClelland and Rumelhart" (https://www.cs.indiana.edu/~port/teach/641/McClellandRumelhart.IAC....), Never knew what was going to happen then & I have seen nothing to convince me it will work any better now!
If I am walking down a mountain and I step into a little pool I stop! This is called hitting a local minimum. For a small number of degrees of freedom (you have 2 above) then you can "effectively" fill the pool with concrete and see if it starts to run over. If it does, you have escaped the local minimum and can keep walking down the mountain. Works OK if you have a small dimensional problem 2-4D, but gets too expensive (time & memory) for more. You can use "simulated annealing", (forget the term, but you need a good pseudorandom number generator). [Say 6D problem, {random pick 1...6, random walk 1...T} repeat ~ T ^1.5, T > 0 and an integer; IF AT ANY POINT, DURING ANY of the random walks, you at a place with a lower value than the current minimum STOP. You have escaped, the local minima. SLOWLY (depends on the problem) reduce the value of T to zero. "T" stand for "temperature" which is where the expression "simulated annealing" comes from. V old method.
Very nicely done. Many years ago, I used the same approach to learn nonparametric statistics using Cohen's textbook. Thanks for presenting this!
Thanks John.
Myself along with a few of my colleagues are starting this online data science training platform where our intent is to actually simplify most of the ML algorithms (like regression, decision trees, GBM, Neural networks and others) through excel sheet demos and case studies.
You can get more information by visiting our website www.deeplearningtrack.com
Great post. Thanks Jahnavi . I also have problems with math notation and I would like to see the same approach applied to other ML concepts.
Hey nice article! And I always like seeing things done in Excel.
Posted 1 March 2021
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