  A scaling law refers to the observation that the test performance of a model
improves as the number of training data increases. A fast scaling law implies
that one can solve machine learning problems by simply boosting the data and
the model sizes. Yet, in many cases, the benefit of adding more data can be
negligible. In this work, we study the rate of scaling laws of nearest neighbor
classifiers. We show that a scaling law can have two phases: in the first
phase, the generalization error depends polynomially on the data dimension and
decreases fast; whereas in the second phase, the error depends exponentially on
the data dimension and decreases slowly. Our analysis highlights the complexity
of the data distribution in determining the generalization error. When the data
distributes benignly, our result suggests that nearest neighbor classifier can
achieve a generalization error that depends polynomially, instead of
exponentially, on the data dimension.
