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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
Oct 25
The following multiplication method makes everybody wish they had been taught math like this in school.

It's not just a cute visual tool: it illuminates how and why long multiplication works.

Here is the full story: Image
First, the method.

The first operand (21 in our case) is represented by two groups of lines: two lines in the first (1st digit), and one in the second (2nd digit).

One group for each digit.
Similarly, the second operand (32) is encoded with two groups of lines, one for each digit.

These lines are perpendicular to the previous ones.
Read 10 tweets
Tivadar Danka Profile picture
Oct 21
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
Oct 20
In calculus, going from a single variable to millions of variables is hard.

Understanding the three main types of functions helps make sense of multivariable calculus.

Surprisingly, they share a deep connection. Let's see why: Image
In general, a function assigns elements of one set to another.

This is too abstract for most engineering applications. Let's zoom in a little! Image
As our measurements are often real numbers, we prefer functions that operate on real vectors or scalars.

There are three categories:

1. vector-scalar,
2. vector-vector,
3. and scalar-vector. Image
Read 16 tweets
Tivadar Danka Profile picture
Oct 19
The Law of Large Numbers is one of the most frequently misunderstood concepts of probability and statistics.

Just because you lost ten blackjack games in a row, it doesn’t mean that you’ll be more likely to be lucky next time.

What is the law of large numbers, then? Read on: Image
The strength of probability theory lies in its ability to translate complex random phenomena into coin tosses, dice rolls, and other simple experiments.

So, let’s stick with coin tossing.

What will the average number of heads be if we toss a coin, say, a thousand times?
To mathematically formalize this question, we’ll need random variables.

Tossing a fair coin is described by the Bernoulli distribution, so let X₁, X₂, … be such independent and identically distributed random variables. Image
Read 17 tweets
Tivadar Danka Profile picture
Oct 15
I have spent at least 50% of my life studying, practicing, and teaching mathematics.

The most common misconceptions I encounter:

• Mathematics is useless
• You must be good with numbers
• You must be talented to do math

These are all wrong. Here's what math is really about: Image
Let's start with a story.

There’s a reason why the best ideas come during showers or walks. They allow the mind to wander freely, unchained from the restraints of focus.

One particular example is graph theory, born from the regular daily walks of the legendary Leonhard Euler.
Here is the map of Königsberg (now known as Kaliningrad, Russia), where these famous walks took place.

This part of the city is interrupted by several rivers and bridges.

(I cheated a little and drew the bridges that were there in Euler's time, but not now). Image
Read 15 tweets
Tivadar Danka Profile picture
Oct 14
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 15 tweets

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