  Reinforcement learning considers the problem of finding policies that
maximize an expected cumulative reward in a Markov decision process with
unknown transition probabilities. In this paper we consider the problem of
finding optimal policies assuming that they belong to a reproducing kernel
Hilbert space (RKHS). To that end we compute unbiased stochastic gradients of
the value function which we use as ascent directions to update the policy. A
major drawback of policy gradient-type algorithms is that they are limited to
episodic tasks unless stationarity assumptions are imposed. Hence preventing
these algorithms to be fully implemented online, which is a desirable property
for systems that need to adapt to new tasks and/or environments in deployment.
The main requirement for a policy gradient algorithm to work is that the
estimate of the gradient at any point in time is an ascent direction for the
initial value function. In this work we establish that indeed this is the case
which enables to show the convergence of the online algorithm to the critical
points of the initial value function. A numerical example shows the ability of
our online algorithm to learn to solve a navigation and surveillance problem,
in which an agent must loop between to goal locations. This example
corroborates our theoretical findings about the ascent directions of subsequent
stochastic gradients. It also shows how the agent running our online algorithm
succeeds in learning to navigate, following a continuing cyclic trajectory that
does not comply with the standard stationarity assumptions in the literature
for non episodic training.
