lin-0056

3.5 A running equivariance sketch

Let a group element \(g\) act on inputs by \(a_g:X\to X\) and on outputs by \(b_g:Y\to Y\). A learned map \(F_\theta :X\to Y\) is declared equivariant by the square

The indexing graph contains the two parallel paths

\[ F_\theta a_g,\qquad b_gF_\theta :X\to Y, \]

and \(\mathcal D\) identifies them. The base obstruction is the failure of the realized diagram to factor through this quotient.

If \(X\) and \(Y\) are normed spaces, one possible observation is

\[ E_{g,\theta }(x) = F_\theta (a_gx)-b_gF_\theta (x), \qquad \ell _{g,\theta }(x)=\lVert E_{g,\theta }(x)\rVert ^2. \]

The vector \(E_{g,\theta }(x)\) and scalar \(\ell _{g,\theta }(x)\) are useful measurements, but neither defines equivariance. The declared square does.

Example 0.17 introduced this square using an image or language encoder. In the language setting, \(a_g\) must preserve or transform explicitly declared structure: it cannot be an arbitrary token permutation of a causal LLM. Depending on the task, \(b_g\) may reindex token-level states or act as the identity on an invariant decision. This choice belongs to the sketch and determines what an observed discrepancy means.

Example 3.6 Bellman consistency

Let \(\mathcal V\) be a value-function space, \(P^\pi :\mathcal V\to \mathcal V\) the Markov expectation operator, and

\[ B^\pi (V)=R^\pi +\gamma P^\pi V \]

the Bellman operator. A parameterized value family \(\widehat V:\Theta \to \mathcal V\) determines the parallel pair

\[ \widehat V,\; B^\pi \widehat V:\Theta \rightrightarrows \mathcal V. \]

Declaring this pair equal expresses Bellman consistency. Sampled TD and GTD objectives are scalar observations of its factorization obstruction; they do not replace the declaration [ Sutton , 1988 , Sutton et al. , 2008 , Sutton and Barto , 2018 ] .