Misconceptions on non-temporal learning: When do machine learning models qualify as prediction systems?

Preamble

    Babylonian Tablet for 
square root of 2.
 (Wikipedia)

Prediction implies a mechanics, as in knowing a form of a trajectory over time.  Strictly speaking a predictive system implies knowing a solution to the path, set of variable depending on time, time evolution of the system under consideration. Here, we define semi-informally how a prediction system is defined mathematically and show how non-temporal learning can be mapped into a prediction system. 

Temporal learning : Recurrence, trajectory and sequences

A trajectory can be seen as a function of time, identified in recurrence manner. It means $x(t_{i})=f(x_{i-1})$. However, this is one of the possible definitions. The physical equivalent of this appears as a solution to ordinary differential equation, such as the velocity $v(t) = dx(t)/dt$, recurrence on its solution. On the other hand machine learning, an empirical approach is taken and a sequence data such as natural language or a log events occurring in sequence. Any modelling on such data is called temporal learning. This includes classical time-series algorithms, gated units in deep learning and differential equations.

Definition: A prediction system  $\mathscr{F}$ that is build with data $D$ but utilised for a data that is not used in building it $D'$, qualified as such if both $D$ and $D'$ are temporal sets and output of the system is a horizon $\mathbb{H}$, that is a sequence. 

Using non-temporal supervised learning is interpolation or extrapolation

Often practice in industry to turn temporal interactions into flat set of data vectors,  $v_{i}$, $i$ corresponds to a time point or an arbitrary property of the dataset resulting in breaking the temporal associations and causal links.  This could also manifest as set of images with some labels which has no ordering or associational property in the dataset. Even though our system build upon these non-temporal datasets, indeed it constituted a learning systems as interpolation or extrapolation. Their utility in using them for $D'$, strictly speaking does not qualify as prediction systems. 

Mapping with pre-processing

A mapping indeed possible from non-temporal data to a temporal one, if their original form is not in temporal form yet. This is been studied in complexity literature. This requires an algorithm to map flattened data vectors we mentioned into a sequence data. 

Mapping with Causality

A distinct models from causal inference are qualified as predictive systems even if they are trained on non-temporal data, because causality establishes a temporal learning.

Non-temporal modals: Do they still learn?

Even though, we exclude non-temporal model utilisation as non-predictive systems, they still classified as learned models. Because their outputs are generated by a learning procedure. 

Conclusion

Differentiation among temporal and non-temporal learning is provided in associational manner. This results into definition of a prediction system, that excludes non-temporal machine learning models: such as models for unlinked set of vectors, a set of numbers mapped from any data modality. 

Further reading & postscript notes

• Practice causal inference: Conventional supervised learning can't do inference
• Causal inference : Editor's selections from the looper repo.
• Causal models usually are not train but validated or so called discovered.