  In real-world control settings, the observation space is often unnecessarily
high-dimensional and subject to time-correlated noise. However, the
controllable dynamics of the system are often far simpler than the dynamics of
the raw observations. It is therefore desirable to learn an encoder to map the
observation space to a simpler space of control-relevant variables. In this
work, we consider the Ex-BMDP model, first proposed by Efroni et al. (2022),
which formalizes control problems where observations can be factorized into an
action-dependent latent state which evolves deterministically, and
action-independent time-correlated noise. Lamb et al. (2022) proposes the
"AC-State" method for learning an encoder to extract a complete
action-dependent latent state representation from the observations in such
problems. AC-State is a multistep-inverse method, in that it uses the encoding
of the the first and last state in a path to predict the first action in the
path. However, we identify cases where AC-State will fail to learn a correct
latent representation of the agent-controllable factor of the state. We
therefore propose a new algorithm, ACDF, which combines multistep-inverse
prediction with a latent forward model. ACDF is guaranteed to correctly infer
an action-dependent latent state encoder for a large class of Ex-BMDP models.
We demonstrate the effectiveness of ACDF on tabular Ex-BMDPs through numerical
simulations; as well as high-dimensional environments using
neural-network-based encoders. Code is available at
https://github.com/midi-lab/acdf.
