  Truncated backpropagation through time (TBPTT) is a popular method for
learning in recurrent neural networks (RNNs) that saves computation and memory
at the cost of bias by truncating backpropagation after a fixed number of lags.
In practice, choosing the optimal truncation length is difficult: TBPTT will
not converge if the truncation length is too small, or will converge slowly if
it is too large. We propose an adaptive TBPTT scheme that converts the problem
from choosing a temporal lag to one of choosing a tolerable amount of gradient
bias. For many realistic RNNs, the TBPTT gradients decay geometrically in
expectation for large lags; under this condition, we can control the bias by
varying the truncation length adaptively. For RNNs with smooth activation
functions, we prove that this bias controls the convergence rate of SGD with
biased gradients for our non-convex loss. Using this theory, we develop a
practical method for adaptively estimating the truncation length during
training. We evaluate our adaptive TBPTT method on synthetic data and language
modeling tasks and find that our adaptive TBPTT ameliorates the computational
pitfalls of fixed TBPTT.
