How Accurate Is Accurate Enough?

Abstract

How accurate must a numerical approximation be within a learning system? Primitive error alone cannot answer this question: errors of the same magnitude can have very different consequences for losses, predictions, and gradients at different learning states. We study this question through the learning objective itself. The objective weights classwise numerical errors nonuniformly according to the current state, so the importance of an error depends not only on its magnitude but also on the class it affects and the weight that class receives. For softmax cross-entropy, we characterize this coupling between class weights and errors and derive the exact extrema of the signed loss change over pairings of fixed non-target probability and score-error multisets, with the target probability and target score error held fixed. Building on this structure, we establish finite-error guarantees that propagate primitive error to losses, probabilities, predictions, and feature gradients, then invert these guarantees to obtain a certified primitive tolerance for the current state under prescribed learning-level error requirements. We give a complete instantiation of the framework in high-dimensional von Mises-Fisher learning. Controlled interventions and a large collection of saved learning states show that identical primitive error can produce substantially different learning consequences, while certified numerical tolerances vary by orders of magnitude across states under the same learning-level requirements. These results show that the adequacy of a numerical approximation must be assessed in relation to the current learning state and the quantity to be preserved; numerical accuracy should itself be treated as part of the learning objective.