TuringUniversal Learners with Optimal Scaling Laws
Abstract
For a given distribution, learning algorithm, and performance metric, the rate of convergence (or datascaling 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 powerlaw 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 distributiondependent 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 bestpossible 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.
 Publication:

arXiv eprints
 Pub Date:
 November 2021
 DOI:
 10.48550/arXiv.2111.05321
 arXiv:
 arXiv:2111.05321
 Bibcode:
 2021arXiv211105321N
 Keywords:

 Computer Science  Machine Learning;
 Computer Science  Artificial Intelligence;
 Computer Science  Computational Complexity;
 Mathematics  Statistics Theory;
 Statistics  Machine Learning