  A common approach for defining a reward function for Multi-objective
Reinforcement Learning (MORL) problems is the weighted sum of the multiple
objectives. The weights are then treated as design parameters dependent on the
expertise (and preference) of the person performing the learning, with the
typical result that a new solution is required for any change in these
settings. This paper investigates the relationship between the reward function
and the optimal value function for MORL; specifically addressing the question
of how to approximate the optimal value function well beyond the set of weights
for which the optimization problem was actually solved, thereby avoiding the
need to recompute for any particular choice. We prove that the value function
transforms smoothly given a transformation of weights of the reward function
(and thus a smooth interpolation in the policy space). A Gaussian process is
used to obtain a smooth interpolation over the reward function weights of the
optimal value function for three well-known examples: GridWorld, Objectworld
and Pendulum. The results show that the interpolation can provide very robust
values for sample states and action space in discrete and continuous domain
problems. Significant advantages arise from utilizing this interpolation
technique in the domain of autonomous vehicles: easy, instant adaptation of
user preferences while driving and true randomization of obstacle vehicle
behavior preferences during training.
