  In the context of supervised learning, meta learning uses features, metadata
and other information to learn about the difficulty, behavior, or composition
of the problem. Using this knowledge can be useful to contextualize classifier
results or allow for targeted decisions about future data sampling. In this
paper, we are specifically interested in learning the Bayes error rate (BER)
based on a labeled data sample. Providing a tight bound on the BER that is also
feasible to estimate has been a challenge. Previous work[1] has shown that a
pairwise bound based on the sum of Henze-Penrose (HP) divergence over label
pairs can be directly estimated using a sum of Friedman-Rafsky (FR)
multivariate run test statistics. However, in situations in which the dataset
and number of classes are large, this bound is computationally infeasible to
calculate and may not be tight. Other multi-class bounds also suffer from
computationally complex estimation procedures. In this paper, we present a
generalized HP divergence measure that allows us to estimate the Bayes error
rate with log-linear computation. We prove that the proposed bound is tighter
than both the pairwise method and a bound proposed by Lin [2]. We also
empirically show that these bounds are close to the BER. We illustrate the
proposed method on the MNIST dataset, and show its utility for the evaluation
of feature reduction strategies. We further demonstrate an approach for
evaluation of deep learning architectures using the proposed bounds.
