good loss in expectation, however the errors are distributed poorly. There might be large errors for states with a small magnitude. The normalized loss on the right allows the large states to have large error and small states to have small error.[]{data-label="fig:model_errs"}](images/model_err.png){width="47.50000%"}

In the unconstrained objective Eq. \[eq:model\_learning\] for model learning, the errors are distributed more or less randomly. States with small magnitudes can have large errors even though, on average, the model error is small. With the sufficiently accurate formulation Eq. \[eq:constrained\_model\_learning\], smaller state magnitudes correspond to smaller errors. This suggests that perhaps the sufficiently accurate model may be easier to use in a controller as small state magnitudes do not have large errors.

To test the effect of the distribution of model errors on controller performance, 500 different Ball-Paddle tasks were generated. Each task varies in goal location of the ball as well as in velocity constraints. Each task is tested with both models using the same controller and results are shown in Table \[table:model\_errs\].

                Unconstrained   Constrained
  ------------ --------------- -------------
    Failure       $20.8 \%$      $0.8 \%$
   Mean error     $0.3136$       $0.2124$

  : **Controller performance with learned models**. A failure is when the ball falls off the paddle, or when it 