What Determines the Optimal Target Transform? Resolution versus Retransformation Bias

Regression models are often trained on a transformed target, such as its logarithm, and their predictions are then mapped back. We study which transform gives the smallest error on the original scale and show that the answer is set by two competing effects. The first is a retransformation bias: mapping predictions back shifts them away from the conditional mean by an amount that grows with the noise in the target, so this effect favors transforms close to the identity. The second is resolution: the transform decides where along the target range the model spends its limited precision, so this effect favors stretching the ranges where most targets lie. We derive an expression that contains these two effects and prove that no rule based on the target distribution alone can pick the best transform for every dataset. In controlled experiments with a fixed target distribution, the best transform moves with the predictability of the target as the analysis predicts, and removing the bias moves it toward stronger compression. Trained models keep their error roughly constant in the units in which they are trained, so standardizing the transformed target matters, but their error is far from uniform across the target range. On 41 tabular datasets, a simple statistic computed from the training targets predicts where the choice of transform matters, the uniform quantile transform is worse than a cube-root transform on every dataset with a neural network, and choosing the transform on a validation set works best in practice.