Nonlinear Label Budgets: How Geometry Determines Prediction Risk

Nonlinear models can exploit low-dimensional structure, but this benefit depends on how observations are processed. We ask how many labels need nonlinear processing when an initial sample configures an affine predictor for the remaining labels. For a known compact smooth model manifold, we characterize the minimax risk over this complete class. The leading risk follows a staircase of derivative-space dimensions as the nonlinear budget changes; at the square-root boundary, curvature determines an exact continuous transition. We construct attaining readouts by correcting the mean displacement and shrinking normal uncertainty. A growing-dimensional construction shows that a vanishing pilot fraction can remove a diverging cost of fixed affine prediction. Allowing a quadratic readout removes the leading curvature penalty with a smaller sufficient pilot budget, while minimax laws quantify source uncertainty and prescribe shrinkage under model error. Conditional low-rank readouts turn these principles into a configuration method with an explicit gain criterion. Experiments test the risk laws and learned readouts; on 100 captured objects, configuration reduces MSE by a per-object average of 16.3% over matched fixed readouts.