  In reinforcement learning (RL), rewards of states are typically considered
additive, and following the Markov assumption, they are $\textit{independent}$
of states visited previously. In many important applications, such as coverage
control, experiment design and informative path planning, rewards naturally
have diminishing returns, i.e., their value decreases in light of similar
states visited previously. To tackle this, we propose $\textit{submodular RL}$
(SubRL), a paradigm which seeks to optimize more general, non-additive (and
history-dependent) rewards modelled via submodular set functions which capture
diminishing returns. Unfortunately, in general, even in tabular settings, we
show that the resulting optimization problem is hard to approximate. On the
other hand, motivated by the success of greedy algorithms in classical
submodular optimization, we propose SubPO, a simple policy gradient-based
algorithm for SubRL that handles non-additive rewards by greedily maximizing
marginal gains. Indeed, under some assumptions on the underlying Markov
Decision Process (MDP), SubPO recovers optimal constant factor approximations
of submodular bandits. Moreover, we derive a natural policy gradient approach
for locally optimizing SubRL instances even in large state- and action- spaces.
We showcase the versatility of our approach by applying SubPO to several
applications, such as biodiversity monitoring, Bayesian experiment design,
informative path planning, and coverage maximization. Our results demonstrate
sample efficiency, as well as scalability to high-dimensional state-action
spaces.
