Backpropagation, backpropagation, backpropagation algorithm, backpropagation
The foundation of how language models work is recognizing relationships between tokens, the smallest units of text. Each word has its place in a multidimensional space of numbers, and based on these relationships, the model can predict subsequent words and create entire sentences.
But how does the model “know” that “cat” is closer to “dog” than to “computer”? Or that the connection between “Germany” and “Bratwurst” resembles the relationship between “Japan” and “sushi”?
There is no magic or manual setup here. At the beginning, all parameters are random; the model understands nothing, and the words are arranged randomly in space. It is pure chaos.
Over time, however, through the learning process, the network begins to discover patterns: it notices that “day” often follows “good,” while “football” usually follows “play.”
How is it possible for understanding of language to emerge from random numbers? The answer lies in three concepts: parameters, tokens, and backpropagation—the foundations of modern artificial intelligence.
Table of contents
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How a language model works (briefly)
A language model predicts the probability of subsequent words based on relationships between tokens, the smallest units of text. Each word has its place in a multidimensional space of numbers—if two words often occur together, their points are close to each other. This enables the model to predict that after “I bought a ball and went to play…” the phrase “football” will most likely appear.
All of the model’s knowledge is encoded in these relationships between words. It contains no ready-made facts or sentences—everything is recorded as mathematical connections between thousands of parameters. Therefore, it is difficult to determine unequivocally what the model “knows.” We cannot find a single place in it that says water boils at 100°C—this information is distributed across the relationships between the words water, heat, and temperature.
To supplement this limited “internal memory,” RAG (Retrieval-Augmented Generation) systems are increasingly used. They allow the model to access external knowledge bases—searching for up-to-date information and incorporating it into its responses. Thanks to this, the model not only understands language but can also refer to real-world data, combining contextual understanding with facts.
What is a token
A token is the smallest unit of text that a model can recognize. Depending on the configuration, it can be:
- the whole word (“house,” “dog”),
- a word fragment (“drank,” “ka”),
- or even a punctuation mark.
The model therefore does not see sentences as a whole—it sees sequences of tokens. Each of them is represented as a vector, that is, a set of numbers describing its “position” in the space of meanings.
At first, this space is random. The coordinates of each token are simply random. The token “ball” has nothing to do with the token “sport.” Only training causes their representations to begin moving closer together in a mathematical sense, which in practice means that the model recognizes the similarity of their meanings.
What Are Parameters?
Every language model has millions, and often billions, of parameters. These are numbers that determine how strong the connection is between individual tokens.
One could say that parameters are the model’s “memory.” They are where all the linguistic patterns, associations, and grammatical structures that the model learned during training are stored.
For example:
- If the model frequently sees a phrase such as “I bought a ball and went to play…”, its parameters will be adjusted so that in the future it predicts that the next word will be something like “football” or “soccer.”
- If “compilation error” often appears in technical sentences, the model will learn that the word “compilation” is related to code and computers, not to sports or cooking.
How the Model Learns Correct Answers
The learning process involves the model being given the task of predicting a missing word in a sentence. This is the fundamental way of training large language models.
Example: “I bought a ball and went to play ___.”
The model must guess what fits in the blank. If it provides “soccer” or “football,” it gets a point. If it chooses “chess,” it is wrong.
This difference between the correct and incorrect answer is called error (loss). It is a signal that tells the model how far it is from the truth.
What is backpropagation?
To improve after an error, the model uses a mechanism called backpropagation. This can be compared to the human learning process: first you try, then you see what went wrong, and finally you adjust your approach.
In the case of a neural network, it looks like this:
- The model makes a prediction (e.g., “chess” instead of “football”).
- The error is calculated (the difference between the prediction and the actual answer).
- The error “flows” backward through the network’s layers, indicating which parameters should be slightly adjusted.
- The neurons’ weights are updated—a tiny step toward a better result.
This cycle is repeated billions of times. As a result, the model’s parameters become increasingly well-fitted to linguistic reality.
The Simplest Analogy: Teacher and Student
Imagine a student learning a new language. They write a sentence: “I go football play.” The teacher says: “No, it should be I play football.” The student remembers this correction.
Something very similar happens in the model: backpropagation acts as a teacher that shows the direction in which the model should improve. But instead of words, it uses equations and gradients—measures of how changing one parameter affects the result.
Although a better example might be archery. After shooting an arrow, I compare the arrow’s distance from the target. If we hit too low, we need to aim higher. How much we raise the bow is precisely the learning rate.
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A Simple Mathematical Example.
Let’s conduct a very simple experiment. Let’s imagine a miniature neural network that needs to learn a simple relationship:
y=2x, i.e.: Output = 2 × input
That is: if we provide 1, we should get 2; if we provide 2, we should get 4, and so on.
Step 1. Initialization
Let:
- w = 0.5 (connection weight, a randomly selected parameter)
- η = 0.1 (learning rate), an arbitrarily selected number. If too low, it makes the model learn too slowly; if too high, the adjustment will be too large, and the model will never learn.
- input x = 1,
- expected output y_true = 2.
At first, the network does not know how it works.
Step 2. Forward Pass
The model calculates the result:
y_pred = x × w = 1 × 0.5 = 0.5
An error is the difference between the result and the truth:
Loss = (y_true − y_pred)² = (2 − 0.5)² = 2.25
Step 3. Backward Pass
Now we need to calculate how to change the weight to reduce the error.
We calculate the derivative of the error with respect to the weight:
dE/dw = 2 × (y_pred − y_true) × x dE/dw = 2 × (0.5 − 2) × 1 = −3
This tells us that reducing the error requires increasing the weight, because the gradient is negative.
Step 4. Updating weight
New value in:
w_new = w − η × dE/dw w_new = 0.5 − 0.1 × (−3) = 0.8
Step 5. Next Prediction
We repeat the calculations:
y_pred = 1 × 0.8 = 0.8 Loss = (2 − 0.8)² = 1.44
The error decreased! Just a dozen or so such iterations are enough for the network to approach the correct solution to ≈ 2.
Mathematical Intuition
Weight update equation:
w ← w − η × ∂E/∂w
This is the essence of learning in artificial intelligence. The symbol ∂E/∂w (read as “the derivative of the error with respect to the weight”) indicates how much changing a given weight affects the total error.
Thanks to this, the model does not guess blindly, but learns in a targeted direction—always toward a smaller error.
Learning in Billions of Dimensions
In the example above, we had only one weight. In reality, language models have billions of weights, and therefore billions of such equations. Each of them is updated after every training step.
In this way, the network creates a vast, multidimensional landscape in which it searches for the lowest point of error—the place where it best captures the patterns in linguistic data.
Backpropagation in Language Model Training
From a mathematical point of view, the model does only one thing: it transforms numbers according to a formula.
From a human point of view, it does something much deeper—it learns meanings.
In the previous example, I used only numbers, making it easier to understand how backpropagation works. In the case of models, words are used instead of numbers (which the program then converts into numbers).
For example, the language model is tasked with completing a sentence, such as: “The cat climbed onto…” If the model uses completely random parameters, it might write, for example: The cat climbed onto … Ecumenism”. The response is then compared with the model answer, namely “The cat climbed onto the tree”; the error is calculated, and the weights are updated.
After billions of iterations, the model begins to understand that:
- the words “ball” and “goal” often occur together,
- “good morning” is a greeting, not a question,
- “because” is usually followed by a reason.
These regularities are not programmed—the model discovers them on its own by analyzing statistics in the data. Each weight adjustment brings it closer to the world of human language.
What Happens to the Tokens
Tokens are the starting point, but after training they become something much more. Their representations (vectors) acquire semantic structure.
In the model space:
- “cat,” “dog,” and “rabbit” are close to one another,
- “table” and “chair” form another group,
- “beautiful,” “ugly,” and “average” lie on a common axis describing aesthetics.
Mathematically, this means that distances between vectors have semantic meaning. The difference between “king” and “queen” is similar to the difference between “man” and “woman” — not because someone programmed it that way, but because the model discovered patterns in the data on its own.
What It Looks Like in Practice
For models such as GPT or Gemini, the training process takes months. Hundreds of thousands of processors analyze vast collections of text: books, articles, websites, and dialogues.
Each token is analyzed in the context of other tokens. The model calculates the probability that the next word will appear after a given sequence of words.
It’s a bit like a student reading the entire internet and, for each sentence, recording what most often comes after what. Backpropagation makes his internal maps of meaning increasingly accurate with every mistake.
Why Does This Work So Well
It is surprising that, despite their mathematical simplicity, backpropagation and learning through context prediction lead to the emergence of something resembling understanding.
The model is not conscious, but it can:
- summarize texts,
- answer questions logically,
- detect contradictions,
- and even write poetry.
This is possible because language is a system full of regularities, and neural networks are masters at discovering regularities hidden in data.
Where Does Mathematics End and Intuition Begin?
It is worth understanding that all these processes—weights, gradients, tokens—are merely tools. Their task is not to “understand” in the human sense, but rather to model probability: which word most often occurs after another.
However, from the user’s perspective, the result is astonishing. A model that started with random numbers can hold a meaningful conversation, explain complex concepts, and adapt its tone to the context.
Summary
- Tokens are the smallest units of language in the model—numbers that represent fragments of text.
- Parameters (weights and biases) determine how strongly neurons respond to these tokens and their interrelationships.
- Backpropagation is a method for adjusting parameters after each mistake.
- The gradient indicates the direction of change—it tells the model how to adjust its parameters to make better predictions.
- Result: the network gradually learns linguistic patterns until it begins to reproduce the structure and meaning of human language.
At first, words are random points in empty space. Then, billion after billion steps, the network begins to see patterns in this chaos. Tokens form groups, meanings connect, and mathematics transforms into something resembling understanding.
This is precisely the magic of backpropagation—a simple differentiation algorithm that makes language emerge from numbers.