  In this paper, we establish a subgame perfect equilibrium reinforcement
learning (SPERL) framework for time-inconsistent (TIC) problems. In the context
of RL, TIC problems are known to face two main challenges: the non-existence of
natural recursive relationships between value functions at different time
points and the violation of Bellman's principle of optimality that raises
questions on the applicability of standard policy iteration algorithms for
unprovable policy improvement theorems. We adapt an extended dynamic
programming theory and propose a new class of algorithms, called backward
policy iteration (BPI), that solves SPERL and addresses both challenges. To
demonstrate the practical usage of BPI as a training framework, we adapt
standard RL simulation methods and derive two BPI-based training algorithms. We
examine our derived training frameworks on a mean-variance portfolio selection
problem and evaluate some performance metrics including convergence and model
identifiability.
