Chapter 4 of 0–21

From scores to probabilities

Logits, the softmax, and how the cross-entropy loss measures wrongness.

The network's final output is not a number like “37” but a list of 113 scores, one per possible answer. This chapter explains how scores become probabilities and how wrongness is measured.

Exponential and logarithm

The exponential function \exp(x) = e^x (with e \approx 2.718) turns any number into a positive one: e^0 = 1, e^1 \approx 2.718, e^{-1} \approx 0.368, e^{5} \approx 148.4. Adding to x multiplies the result: e^{x+1} = e\cdot e^x.

The natural logarithm undoes it: \ln(e^x) = x. So \ln 1 = 0, \ln 113 \approx 4.727, and \ln of a number between 0 and 1 is negative (\ln 0.5 \approx -0.693).

From scores to a probability distribution

A probability distribution over the 113 answers is a list of 113 non-negative numbers that add up to 1. The network instead outputs one unconstrained score per answer — a logit. The softmax function converts logits into probabilities:

Exponentials make every score positive; dividing by their total makes them add up to 1. Adding the same number to all logits changes nothing (it multiplies numerator and denominator by the same factor) — only differences between logits matter.

Press ▶ next to “scale all logits” to multiply every logit by a growing factor: the probabilities pile up on the largest logit, so the loss shrinks when the correct answer has the largest logit and grows when it does not.

The predicted answer is the argmax — the position of the largest logit (equivalently, of the largest probability). With only two options, softmax has a simple form: the probability of the first is \frac{e^{x_1}}{e^{x_1}+e^{x_2}} = \frac{1}{1+e^{-(x_1-x_2)}}, the sigmoid \sigma of the difference. We will meet this when an attention head chooses between a and b.

Measuring wrongness: the cross-entropy loss

A loss is a single number saying how wrong the outputs are. The network is trained with the cross-entropy loss loss: for one example it is -\ln p, where p is the probability given to the correct answer, and for a set of examples we average [28] [33]:

The fraction of examples whose argmax is the correct answer is the accuracy. Accuracy only cares whether the right answer is ranked first; loss also cares how confident the network is. The paper stresses a consequence:

“Zero loss does not exist with cross-entropy loss.”— Nanda et al. 2023, footnote 4, p. 33 · source

Even with 100 % accuracy, the loss can always be lowered by making the logits more extreme, so training never “finishes” — which will matter when we look at what weight decay does in chapter 19.