lin-0052

3.1 From computational graphs to learning sketches

Let \(S\) be a directed graph of formal learning operations and let

\[ J=\operatorname {Path}(S) \]

be its free category. Objects of \(J\) can denote data, representations, predictions, policies, local sections, or world-model fragments. Its arrows are formal composites of encoders, transitions, interventions, restrictions, updates, and decision maps.

A bare graph says which composites can be formed. It does not say which composites ought to agree. A learning sketch adds that declaration.

Definition 3.1 Learning sketch

A learning sketch is a quadruple

\[ \mathbb S=(S,\mathcal D,\mathcal L,\mathcal K), \]

where:

  • \(S\) is a graph of formal learning operations;

  • \(\mathcal D\) is a family of equations between parallel paths;

  • \(\mathcal L\) is a family of designated cones; and

  • \(\mathcal K\) is a family of designated cocones.

The equations in \(\mathcal D\) generate a congruence \(\sim _{\mathcal D}\) on \(J=\operatorname {Path}(S)\) and hence a quotient

\[ q_{\mathcal D}:J\longrightarrow J/{\sim _{\mathcal D}}. \]

The three kinds of declaration play different roles. Path equations express route agreement. Designated cones express reconstruction, synchronization, or shared-state conditions. Designated cocones express aggregation, identification, or gluing. Treating all three as a single scalar penalty would erase this distinction.

Design principle

Declare route equalities and universal properties separately. They can fail for different reasons, localize to different components, and require different repair languages.