  In reinforcement learning, temporal difference (TD) is the most direct
algorithm to learn the value function of a policy. For large or infinite state
spaces, exact representations of the value function are usually not available,
and it must be approximated by a function in some parametric family.
  However, with \emph{nonlinear} parametric approximations (such as neural
networks), TD is not guaranteed to converge to a good approximation of the true
value function within the family, and is known to diverge even in relatively
simple cases. TD lacks an interpretation as a stochastic gradient descent of an
error between the true and approximate value functions, which would provide
such guarantees.
  We prove that approximate TD is a gradient descent provided the current
policy is \emph{reversible}. This holds even with nonlinear approximations.
  A policy with transition probabilities $P(s,s')$ between states is reversible
if there exists a function $\mu$ over states such that
$\frac{P(s,s')}{P(s',s)}=\frac{\mu(s')}{\mu(s)}$. In particular, every move can
be undone with some probability. This condition is restrictive; it is
satisfied, for instance, for a navigation problem in any unoriented graph.
  In this case, approximate TD is exactly a gradient descent of the
\emph{Dirichlet norm}, the norm of the difference of \emph{gradients} between
the true and approximate value functions. The Dirichlet norm also controls the
bias of approximate policy gradient. These results hold even with no decay
factor ($\gamma=1$) and do not rely on contractivity of the Bellman operator,
thus proving stability of TD even with $\gamma=1$ for reversible policies.

Machine Learning;Optimization and Control;Machine Learning