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As modern machine learning models continue to advance the computationalfrontier, it has become increasingly important to develop precise estimates forexpected performance improvements under different model and data scalingregimes. Currently, theoretical understanding of the learning curves thatcharacterize how the prediction error depends on the number of samples isrestricted to either large-sample asymptotics ($m\to\infty$) or, for certainsimple data distributions, to the high-dimensional asymptotics in which thenumber of samples scales linearly with the dimension ($m\propto d$). There is awide gulf between these two regimes, including all higher-order scalingrelations $m\propto d^r$, which are the subject of the present paper. We focuson the problem of kernel ridge regression for dot-product kernels and presentprecise formulas for the mean of the test error, bias, and variance, for datadrawn uniformly from the sphere with isotropic random labels in the $r$th-orderasymptotic scaling regime $m\to\infty$ with $m/d^r$ held constant. We observe apeak in the learning curve whenever $m \approx d^r/r!$ for any integer $r$,leading to multiple sample-wise descent and nontrivial behavior at multiplescales.

Precise Learning Curves and Higher-Order Scaling Limits for Dot Product Kernel Regression