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

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. Email addresses: michael.fairbank.1 'at' city.ac.uk (Michael Fairbank), eduardo'at' soi.city.ac.uk (Eduardo Alonso) Preprint submitted to Arxiv January 8, 2018 1. Introduction Reinforcement learning (RL) is the study of how an agent can learn actions that maximise some given reward function. The robot keeps moving until it reaches one of the designated terminal states. One key approach to tackle this RL problem is to assign a score to every point in state space that gives the best possible total reward attainable if starting from that state.