  We present a framework for compactly summarizing many recent results in
efficient and/or biologically plausible online training of recurrent neural
networks (RNN). The framework organizes algorithms according to several
criteria: (a) past vs. future facing, (b) tensor structure, (c) stochastic vs.
deterministic, and (d) closed form vs. numerical. These axes reveal latent
conceptual connections among several recent advances in online learning.
Furthermore, we provide novel mathematical intuitions for their degree of
success. Testing various algorithms on two synthetic tasks shows that
performances cluster according to our criteria. Although a similar clustering
is also observed for gradient alignment, alignment with exact methods does not
alone explain ultimate performance, especially for stochastic algorithms. This
suggests the need for better comparison metrics.
