  Over the past decade there has been considerable interest in spectral
algorithms for learning Predictive State Representations (PSRs). Spectral
algorithms have appealing theoretical guarantees; however, the resulting models
do not always perform well on inference tasks in practice. One reason for this
behavior is the mismatch between the intended task (accurate filtering or
prediction) and the loss function being optimized by the algorithm (estimation
error in model parameters).
  A natural idea is to improve performance by refining PSRs using an algorithm
such as EM. Unfortunately it is not obvious how to apply apply an EM style
algorithm in the context of PSRs as the Log Likelihood is not well defined for
all PSRs. We show that it is possible to overcome this problem using ideas from
Predictive State Inference Machines.
  We combine spectral algorithms for PSRs as a consistent and efficient
initialization with PSIM-style updates to refine the resulting model
parameters. By combining these two ideas we develop Inference Gradients, a
simple, fast, and robust method for practical learning of PSRs. Inference
Gradients performs gradient descent in the PSR parameter space to optimize an
inference-based loss function like PSIM. Because Inference Gradients uses a
spectral initialization we get the same consistency benefits as PSRs. We show
that Inference Gradients outperforms both PSRs and PSIMs on real and synthetic
data sets.
