  For a given distribution, learning algorithm, and performance metric, the
rate of convergence (or data-scaling law) is the asymptotic behavior of the
algorithm's test performance as a function of number of train samples. Many
learning methods in both theory and practice have power-law rates, i.e.
performance scales as $n^{-\alpha}$ for some $\alpha > 0$. Moreover, both
theoreticians and practitioners are concerned with improving the rates of their
learning algorithms under settings of interest. We observe the existence of a
"universal learner", which achieves the best possible distribution-dependent
asymptotic rate among all learning algorithms within a specified runtime (e.g.
$O(n^2)$), while incurring only polylogarithmic slowdown over this runtime.
This algorithm is uniform, and does not depend on the distribution, and yet
achieves best-possible rates for all distributions. The construction itself is
a simple extension of Levin's universal search (Levin, 1973). And much like
universal search, the universal learner is not at all practical, and is
primarily of theoretical and philosophical interest.
