Maximum Likelihood

Pick the parameters that make the data you actually observed as probable as possible. It is the answer to "how should I fit this?" that almost everything in the corpus is an instance of — and once you see it, cross-entropy, next-token prediction and most of the loss functions here stop looking like separate inventions.

Training a language model to predict the next token IS maximum likelihood, with the corpus as the data and the softmax as the model. So is fitting a linear regression under Gaussian noise, which is where least squares comes from rather than from anywhere geometric. Recognising this is practical rather than decorative: it tells you what a loss assumes about the noise, and a loss whose implied assumption is wrong for your data will be confidently wrong in a specific direction.

Maximising the likelihood of independent observations means maximising a product, which underflows immediately, so the log is taken and the product becomes a sum — and maximising that sum is minimising its negative, which is cross-entropy. The three names describe one quantity. The estimator is consistent and asymptotically efficient under its assumptions, and it has no penalty for complexity whatsoever: it will fit noise exactly if the model can, which is why every practical version adds something the principle itself does not contain.