Asymptotics of Ridge(less) Regression under General Source Condition

06/11/2020

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by   Dominic Richards, et al.

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We analyze the prediction performance of ridge and ridgeless regression when both the number and the dimension of the data go to infinity. In particular, we consider a general setting introducing prior assumptions characterizing "easy" and "hard" learning problems. In this setting, we show that ridgeless (zero regularisation) regression is optimal for easy problems with a high signal to noise. Furthermore, we show that additional descents in the ridgeless bias and variance learning curve can occur beyond the interpolating threshold, verifying recent empirical observations. More generally, we show how a variety of learning curves are possible depending on the problem at hand. From a technical point of view, characterising the influence of prior assumptions requires extending previous applications of random matrix theory to study ridge regression.