  In this theoretical paper we are concerned with the problem of learning a
value function by a smooth general function approximator, to solve a
deterministic episodic control problem in a large continuous state space. It is
shown that learning the gradient of the value-function at every point along a
trajectory generated by a greedy policy is a sufficient condition for the
trajectory to be locally extremal, and often locally optimal, and we argue that
this brings greater efficiency to value-function learning. This contrasts to
traditional value-function learning in which the value-function must be learnt
over the whole of state space.
  It is also proven that policy-gradient learning applied to a greedy policy on
a value-function produces a weight update equivalent to a value-gradient weight
update, which provides a surprising connection between these two alternative
paradigms of reinforcement learning, and a convergence proof for control
problems with a value function represented by a general smooth function
approximator.
