  Upside-Down Reinforcement Learning (UDRL) is an approach for solving RL
problems that does not require value functions and uses only supervised
learning, where the targets for given inputs in a dataset do not change over
time. Ghosh et al. proved that Goal-Conditional Supervised Learning (GCSL) --
which can be viewed as a simplified version of UDRL -- optimizes a lower bound
on goal-reaching performance. This raises expectations that such algorithms may
enjoy guaranteed convergence to the optimal policy in arbitrary environments,
similar to certain well-known traditional RL algorithms. Here we show that for
a specific episodic UDRL algorithm (eUDRL, including GCSL), this is not the
case, and give the causes of this limitation. To do so, we first introduce a
helpful rewrite of eUDRL as a recursive policy update. This formulation helps
to disprove its convergence to the optimal policy for a wide class of
stochastic environments. Finally, we provide a concrete example of a very
simple environment where eUDRL diverges. Since the primary aim of this paper is
to present a negative result, and the best counterexamples are the simplest
ones, we restrict all discussions to finite (discrete) environments, ignoring
issues of function approximation and limited sample size.
