  Representation learning is a fundamental task in machine learning, aiming at
uncovering structures from data to facilitate subsequent tasks. However, what
is a good representation for planning and reasoning in a stochastic world
remains an open problem. In this work, we posit that learning a distance
function is essential to allow planning and reasoning in the representation
space. We show that a geometric abstraction of the probabilistic world dynamics
can be embedded into the representation space through asymmetric contrastive
learning. Unlike previous approaches that focus on learning mutual similarity
or compatibility measures, we instead learn an asymmetric similarity function
that reflects the state reachability and allows multi-way probabilistic
inference. Moreover, by conditioning on a common reference state (e.g. the
observer's current state), the learned representation space allows us to
discover the geometrically salient states that only a handful of paths can lead
through. These states can naturally serve as subgoals to break down
long-horizon planning tasks. We evaluate our method in gridworld environments
with various layouts and demonstrate its effectiveness in discovering the
subgoals.
