  Methods for learning from data depend on various types of tuning parameters,
such as penalization strength or step size. Since performance can depend
strongly on these parameters, it is important to compare classes of
estimators-by considering prescribed finite sets of tuning parameters-not just
particularly tuned methods. In this work, we investigate classes of methods via
the relative performance of the best method in the class. We consider the
central problem of linear regression-with a random isotropic ground truth-and
investigate the estimation performance of two fundamental methods, gradient
descent and ridge regression. We unveil the following phenomena. (1) For
general designs, constant stepsize gradient descent outperforms ridge
regression when the eigenvalues of the empirical data covariance matrix decay
slowly, as a power law with exponent less than unity. If instead the
eigenvalues decay quickly, as a power law with exponent greater than unity or
exponentially, we show that ridge regression outperforms gradient descent. (2)
For orthogonal designs, we compute the exact minimax optimal class of
estimators (achieving min-max-min optimality), showing it is equivalent to
gradient descent with decaying learning rate. We find the sub-optimality of
ridge regression and gradient descent with constant step size. Our results
highlight that statistical performance can depend strongly on tuning
parameters. In particular, while optimally tuned ridge regression is the best
estimator in our setting, it can be outperformed by gradient descent by an
arbitrary/unbounded amount when both methods are only tuned over finitely many
regularization parameters.
