decision rule (argmax prediction)

The decision rule is the final, often overlooked, step that turns a model's continuous output scores into a single discrete prediction. For standard single-label classification this rule is argmax: among the C class scores, predict the class with the largest value. Because softmax is monotonic, taking the argmax of the logits and of the softmax probabilities gives the same class, so for a plain top-1 prediction you do not even need to compute the softmax.

Argmax is the optimal rule under a specific, usually unstated assumption: that every type of error costs the same, a symmetric 0/1 loss, and that you simply want to maximise accuracy. Under those conditions, predicting the most probable class minimises expected error. The moment that assumption breaks, argmax is no longer optimal, and recognising this is what separates a careful practitioner from a naive one.

When errors have unequal costs, for instance missing a tumour is worse than a false alarm, or when class priors at deployment differ from training, or when you must abstain on low-confidence inputs, you replace plain argmax with a thresholded, cost-weighted, or reject-option rule. For multi-label problems argmax is wrong outright, and you threshold each independent sigmoid instead. The decision rule is thus a deliberate design choice, not an automatic afterthought.