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Tivadar Danka Profile picture
@TivadarDanka
Oct 5, 2021 • 10 tweets • 4 min read • Read on X
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There is one big reason we love the logarithm function in machine learning.

Logarithms help us reduce complexity by turning multiplication into addition. You might not know it, but they are behind a lot of things in machine learning.

Here is the entire story.

🧵 👇🏽
First, let's start with the definition of the logarithm.

The base 𝑎 logarithm of 𝑏 is simply the solution of the equation 𝑎ˣ = 𝑏.

Despite its simplicity, it has many useful properties that we take advantage of all the time.
You can think of the logarithm as the inverse of exponentiation.

Because of this, it turns multiplication into addition. Exponentiation does the opposite: it turns addition into multiplication.

(The base is often assumed to be a fixed constant. Thus, it can be omitted.)
Why is this useful? For calculating gradients and derivatives!

Training a neural network requires finding its gradient. However, lots of commonly used functions are written in terms of products.

As you can see, this complicates things.
By taking the logarithm, we can compute the derivative as it turns products into sums.

This method is called logarithmic differentiation.

Since the logarithm is increasing, maximizing a function is the same as maximizing its logarithm. (Same with minimization.)
One example where this is useful is the maximum likelihood estimation.

Given a set of observations and a predictive model, we can write this in the following form.
Believe it or not, this is behind the mean squared error.

Every time you use this, logarithms are working in the background.
If you enjoyed this thread and want to see behind the curtain of machine learning, I am writing a book for you, where we go from high school math to neural networks, one step at a time.

The early access for Mathematics of Machine Learning is out now!

tivadar.gumroad.com/l/mathematics-…
More applications of logarithms: transforming data for visualization. This is extremely useful in life sciences, where the scale of features is exponential.

And one more: numerical stability. When a computation involves very large numbers, taking the log can help avoid overflow.

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More from @TivadarDanka

Tivadar Danka Profile picture
Sep 11
Logistic regression is one of the simplest models in machine learning, and one of the most revealing.

It shows how to move from geometric intuition to probabilistic reasoning. Mastering it sets the foundation for everything else.

Let’s dissect it step by step! Image
Let’s start with the most basic setup possible: one feature, two classes.

You’re predicting if a student passes or fails based on hours studied.

Your input x is a number, and your output y is either 0 or 1.

Let's build a predictive model! Image
We need a model that outputs values between 0 and 1.

Enter the sigmoid function: σ(ax + b).

If σ(ax + b) > 0.5, we predict pass (1).

Otherwise, fail (0).

It’s a clean way to represent uncertainty with math. Image
Read 15 tweets
Tivadar Danka Profile picture
Sep 8
Matrix multiplication is not easy to understand.

Even looking at the definition used to make me sweat, let alone trying to comprehend the pattern. Yet, there is a stunningly simple explanation behind it.

Let's pull back the curtain! Image
First, the raw definition.

This is how the product of A and B is given. Not the easiest (or most pleasant) to look at.

We are going to unwrap this. Image
Here is a quick visualization before the technical details.

The element in the i-th row and j-th column of AB is the dot product of A's i-th row and B's j-th column. Image
Read 16 tweets
Tivadar Danka Profile picture
Sep 7
Behold one of the mightiest tools in mathematics: the camel principle.

I am dead serious. Deep down, this tiny rule is the cog in many methods. Ones that you use every day.

Here is what it is, how it works, and why it is essential: Image
First, the story:

The old Arab passes away, leaving half of his fortune to his eldest son, third to his middle son, and ninth to his smallest.

Upon opening the stable, they realize that the old man had 17 camels. Image
This is a problem, as they cannot split 17 camels into 1/2, 1/3, and 1/9 without cutting some in half.

So, they turn to the wise neighbor for advice. Image
Read 18 tweets
Tivadar Danka Profile picture
Sep 7
The way you think about the exponential function is wrong.

Don't think so? I'll convince you. Did you realize that multiplying e by itself π times doesn't make sense?

Here is what's really behind the most important function of all time: Image
First things first: terminologies.

The expression aᵇ is read "a raised to the power of b."

(Or a to the b in short.) Image
The number a is called the base, and b is called the exponent.

Let's start with the basics: positive integer exponents. By definition, aⁿ is the repeated multiplication of a by itself n times.

Sounds simple enough. Image
Read 18 tweets
Tivadar Danka Profile picture
Sep 5
In machine learning, we use the dot product every day.

However, its definition is far from revealing. For instance, what does it have to do with similarity?

There is a beautiful geometric explanation behind: Image
By definition, the dot product (or inner product) of two vectors is defined by the sum of coordinate products. Image
To peek behind the curtain, there are three key properties that we have to understand.

First, the dot product is linear in both variables. This property is called bilinearity. Image
Read 16 tweets
Tivadar Danka Profile picture
Sep 5
The single biggest argument about statistics: is probability frequentist or Bayesian?

It's neither, and I'll explain why.

Deep-dive explanation incoming: Image
First, let's look at what probability is.

Probability quantitatively measures the likelihood of events, like rolling six with a die. It's a number between zero and one.

This is independent of interpretation; it’s a rule set in stone. Image
In the language of probability theory, the events are formalized by sets within an event space.

The event space is also a set, usually denoted by Ω.) Image
Read 34 tweets

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