LLM Pre-training

July 11, 2026 · Updated September 24, 2026

Contents

Pretraining teaches the model what the world and language look like.

Suppose the training text is “The capital of France is Paris.” This gets preprocessed into training examples:

Input: “The” → Target: “capital”
Input: “The capital” → Target: “of”
Input: “The capital of” → Target: “France”

The loss is

L = -\sum_t \log p_\theta(x_t \mid x_{<t})

To see where this comes from, suppose the training text is “The cat sat.” The model sees:

"The"     → predict "cat"
"The cat" → predict "sat"

At each position, the model outputs a probability distribution over all possible next tokens. For example:

Context	True next token	Model probability
”The”	cat	0.9
”The cat”	sat	0.8

The probability the model assigns to the entire sequence is

p_\theta(x_1, x_2, \dots, x_T) = \prod_{t=1}^T p_\theta(x_t \mid x_{<t})

Using the chain rule,

p(\text{"The cat sat"}) = p(\text{cat} \mid \text{The}) \times p(\text{sat} \mid \text{The cat}) = 0.9 \times 0.8 = 0.72

Pretraining tries to make the training text as probable as possible: maximize $p_\theta(\text{training data})$ .

Why we take logs

Since the probability of an entire sequence is a product,

p = p(x_1)\, p(x_2 \mid x_1)\, p(x_3 \mid x_1, x_2) \cdots

this is inconvenient: derivatives are messy, and probabilities become tiny very quickly. Instead we take the logarithm. Since $\log(ab) = \log a + \log b$ ,

\log p(\text{sequence}) = \sum_t \log p(x_t \mid x_{<t})

and we optimize a sum instead of a product. This helps for two reasons:

Numerical stability: products of many probabilities quickly underflow to zero in floating-point arithmetic, while sums of logs remain well-behaved.
Optimization: gradients of the log likelihood are much cleaner.

Taking the log doesn’t change the objective, because the logarithm is monotonically increasing: if $a > b$ then $\log a > \log b$ , so the sequence with the highest probability also has the highest log probability.

From log-likelihood to loss

Maximizing $f$ is mathematically identical to minimizing $-f$ , and frameworks like PyTorch, TensorFlow, and JAX expect you to provide a scalar loss and then compute its gradient to minimize it. So instead of “maximize log likelihood” we say “minimize negative log likelihood”; they are the same optimization problem. So instead of

\max_\theta \sum_t \log p_\theta(x_t \mid x_{<t})

we define

L = -\sum_t \log p_\theta(x_t \mid x_{<t})

Suppose the true next token is “cat”:

Good prediction: the model says cat: 0.99. Loss contribution: $-\log(0.99) \approx 0.01$ , a tiny penalty.
Decent prediction: the model says cat: 0.6. Loss: $-\log(0.6) \approx 0.51$ , a bigger penalty.

TODO

“GPT Pretraining Explained: From Raw Text to Next-Token Prediction”, TensorTonic
MiMo: Unlocking the Reasoning Potential of Language Model, From Pretraining to Posttraining
Scaling laws: CS336 Lecture 9 (PDF).
CS336 Assignment 3: Scaling. Understand the function of each component of the Transformer, and query a training API to fit a scaling law that projects how a model scales.
CS336 Assignment 4: Data. Convert raw Common Crawl dumps into usable pretraining data, filtering and deduplicating it to improve model performance.