Field crosswalk — CIRL / Scalar Reward Inference

Cooperative inverse reinforcement learning treats the inferred object as a scalar reward. On the book's shared finite domain, that is exactly the k=1 bundle case; full bundle transport implies cooperative readability, but scalar inference does not determine bundle geometry.

What decision changes?

When evaluating CIRL-style reward learning, ask whether the target is a single reward coordinate or bundle structure, bearer maps, and the correction process that produces later evidence.

The field object is cooperative inverse reinforcement learning: a human knows a reward (or latent type); the robot infers it from observations and acts to maximize cooperative return. Scalar CIRL collapses the inferred object to one coordinate.

The book’s stronger target is bundle inference — value directions, bearer maps, and transport that survive transformation. Lean proves the honest special case: scalar assistance games embed as bundle games with k=1, and full transport implies cooperative readability. It also proves the separation: cooperative scalar inference can hold while bundle geometry fails to preserve across profiles that share the same scalar marginal.

What CIRL keeps that this crosswalk does not replace: a concrete online assistance-game protocol and a training story for how the reward is actually learned. The book reframes the target, not the learning algorithm.

Formulas

ScalarCIRL(p,r)    BundleInf(p,rˉ),rˉ(a)=r for all actions a\text{ScalarCIRL}(p,r) \iff \text{BundleInf}(p,\,\bar r),\quad \bar r(a)=r\ \text{for all actions }a
Scalar cooperative inference embeds as one-bundle-coordinate bundle inference: the scalar reward r is lifted to a constant bundle coordinate. (ch16)
FullTransport(A,B)  CooperativeRewardInference(A,B)\text{FullTransport}(A,B)\ \Rightarrow\ \text{CooperativeRewardInference}(A,B)
Forward projection: preserving full bundle/bearer/correction transport implies the weaker CIRL-readable semantics layer. (ch46)
CooperativeRewardInference(p,r)  ¬ValueBundleGeometryPreserved(p,p)\text{CooperativeRewardInference}(p,r)\ \wedge\ \neg\text{ValueBundleGeometryPreserved}(p,p')
Non-converse separation: the same scalar marginal and policy can hide different bundle geometry — scalar inference is strictly weaker than geometry preservation. (ch46)