3. Loss and how wrong we are

To learn, the network needs a number telling it how badly it’s doing. That’s the loss function.

For language models the task is: given the text so far, predict the next token. The network outputs a probability for every possible token. If the true next token was “cat” and the model gave “cat” a probability of 0.02, that’s bad. If it gave 0.9, that’s good.

The loss used is cross-entropy (also called negative log-likelihood):

$L = - lo g p (correct token)$

Averaged over a whole text of $N$ tokens, this becomes:

$L = - \frac{1}{N} \sum_{t = 1}^{N} lo g p_{θ} (x_{t} ∣ x_{< t})$

Breaking down the averaged formula, piece by piece

Piece How to read it
$\sum_{t = 1}^{N}$ ”For every position $t$ from 1 to $N$ , add up…”
$x_{t}$ The actual token at position $t$
$x_{< t}$ All the tokens before position $t$ (the context the model gets to see)
$p_{θ} (x_{t} ∣ x_{< t})$ “The probability the model (with parameters $θ$ ) assigned to the true token, given everything before it”
$lo g (\cdot)$ Natural log. Turns a probability into a negative number (or 0 if perfect)
$-$ (leading minus) Flips it so the loss is positive. Bigger = worse
$\frac{1}{N}$ Divide by the count, i.e. take the average

Plain English: “On average, how surprised was the model by the real next token?”

Piece	How to read it
$\sum_{t = 1}^{N}$	”For every position $t$ from 1 to $N$ , add up…”
$x_{t}$	The actual token at position $t$
$x_{< t}$	All the tokens before position $t$ (the context the model gets to see)
$p_{θ} (x_{t} ∣ x_{< t})$	“The probability the model (with parameters $θ$ ) assigned to the true token, given everything before it”
$lo g (\cdot)$	Natural log. Turns a probability into a negative number (or 0 if perfect)
$-$ (leading minus)	Flips it so the loss is positive. Bigger = worse
$\frac{1}{N}$	Divide by the count, i.e. take the average

Model’s probability for the right token	Loss $- lo g p$	Reaction
$p = 1.0$	$0$	Perfect
$p = 0.9$	$\approx 0.11$	Great
$p = 0.5$	$\approx 0.69$	Meh
$p = 0.02$	$\approx 3.9$	Bad
$p = 0.01$	$\approx 4.6$	Very bad
$p \to 0$	$\to \infty$	Catastrophic

The log makes confident wrong answers get punished hard. Training means: adjust the parameters so this loss, averaged over a gigantic amount of text, goes down.

Perplexity

You’ll also see perplexity:

$perplexity = e^{L}$

Roughly “how many tokens is the model effectively choosing between at each step.” A perplexity of 10 means it’s about as unsure as picking uniformly from 10 options.

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3. Loss and how wrong we are

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