7 Infinitesimal Decisions

The Tangent Layer of Universal Decision Learning

Infinitesimal causality begins with a grounded intervention protocol and asks how its local effects compose. Decision making begins with a different problem: partial information is available in some contexts, yet an agent must act coherently in contexts it has not directly encountered. Universal Decision Learning (UDL) formulates this passage from local information to global behavior through left and right Kan extensions [ Mahadevan , 2026l ] .

This chapter develops the tangent abstraction of that construction. An infinitesimal decision is not a small action. It is the first-order change in a universally extended decision semantics under an admitted perturbation of its data, context, constraints, or model. The resulting calculus applies to planning, constrained optimization, games, causal decisions, and online learning as well as reinforcement learning. GIRL, developed later in Chapter 15, is the Bellman and policy specialization of this more general layer.

¹1. The word decision refers to the entire semantics that generates and tests admissible behavior, not merely to a terminal discrete action. Its tangent may expose changes in candidates, constraints, values, equilibria, or policies. ↩