sec-comparator-spectrum

12.2 The comparator spectrum on a finite horizon

Let \([T]\) be a finite chain and partition it into nonempty contiguous blocks

\[ B_1{\lt}\cdots {\lt}B_m. \]

Write \(q_B:[T]\to [m]\) for the monotone map sending a time to its block.

Proposition 12.4 Block-static comparator representation

For every category \(\mathcal D\), precomposition

\[ q_B^\ast :[[m],\mathcal D]\longrightarrow [[T],\mathcal D] \]

is fully faithful. If the pointwise right Kan extension exists, a trajectory \(c:[T]\to \mathcal D\) is in its essential image exactly when every transition \(c(s\le t)\) with \(s,t\in B_j\) is an isomorphism.

Proof ▼

A natural transformation between two pulled-back diagrams is constant on each block and is determined uniquely by its components at one representative of every block. Naturality on \([T]\) supplies exactly the naturality conditions on \([m]\), so \(q_B^\ast \) is fully faithful.

If \(c\cong q_B^\ast r\), all within-block arrows are identities up to the displayed natural isomorphism. Conversely, choose the first time \(b_j\) in each block and define \(r(j)=c(b_j)\), with transition maps inherited from the unique arrows of \([T]\). When all within-block transitions are invertible, the maps \(c(b_j\le t)\) give a natural isomorphism \(q_B^\ast r\cong c\).

Refinement of partitions now orders comparator power. If \(B'\) refines \(B\), there is a unique monotone map \(u\) with

\[ q_B=u\circ q_{B'}, \qquad q_B^\ast =q_{B'}^\ast u^\ast . \]

Hence every \(B\)-block-static comparator is also \(B'\)-block-static:

\[ \mathsf{Comp}_{\mathrm{static}} \subseteq \mathsf{Comp}_{B} \subseteq \mathsf{Comp}_{B'} \subseteq \mathsf{Comp}_{\mathrm{dynamic}}. \]

The endpoints are the one-block partition and the singleton partition. This is an information-regret spectrum generated by refinement of reference shape, not by changing the learner.

Corollary 12.5 Monotonicity of scalar comparator regret

In a real-valued minimization problem, let \(\mathcal C\subseteq \mathcal C'\) be comparator classes induced by a sketch refinement. Whenever both infima exist,

\[ \inf _{c'\in \mathcal C'}L(c') \le \inf _{c\in \mathcal C}L(c), \]

and therefore

\[ L^{\mathrm{on}}-\inf _{c\in \mathcal C}L(c) \le L^{\mathrm{on}}-\inf _{c'\in \mathcal C'}L(c'). \]

A stronger comparator can only increase the resulting external-regret number. The inclusion of comparator objects is universal; the inequalities appear only after applying the ordered numerical observer.