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In this theoretical paper we are concerned with the problem of learning avalue function by a smooth general function approximator, to solve adeterministic episodic control problem in a large continuous state space. It isshown that learning the gradient of the value-function at every point along atrajectory generated by a greedy policy is a sufficient condition for thetrajectory to be locally extremal, and often locally optimal, and we argue thatthis brings greater efficiency to value-function learning. This contrasts totraditional value-function learning in which the value-function must be learntover the whole of state space.  It is also proven that policy-gradient learning applied to a greedy policy ona value-function produces a weight update equivalent to a value-gradient weightupdate, which provides a surprising connection between these two alternativeparadigms of reinforcement learning, and a convergence proof for controlproblems with a value function represented by a general smooth functionapproximator.

The Local Optimality of Reinforcement Learning by Value Gradients, and its Relationship to Policy Gradient Learning