In situations like that, you get into an optimized fixed point over time, even though the learning algorithm itself isn’t explicitly searching for that.
Note, if the prediction algorithm anticipates this process (perhaps partially), it will “jump ahead”, so that convergence to a fixed point happens more within the computation of the predictor (less over steps of real world interaction). This isn’t formally the same as searching for fixed points internally (you will get much weaker guarantees out of this haphazard process), but it does mean optimization for fixed point finding is happening within the system under some conditions.
To highlight the “blurry distinction” more:
Note, if the prediction algorithm anticipates this process (perhaps partially), it will “jump ahead”, so that convergence to a fixed point happens more within the computation of the predictor (less over steps of real world interaction). This isn’t formally the same as searching for fixed points internally (you will get much weaker guarantees out of this haphazard process), but it does mean optimization for fixed point finding is happening within the system under some conditions.