lin-0099

7.6 Nonsmooth and set-valued decisions

Decision systems are often nonsmooth precisely where the interesting decision changes. An \(\arg \max \) can switch branches, a ReLU policy can cross an activation boundary, and a constrained optimum can acquire a new active constraint. Requiring a unique classical derivative at every such point would discard rather than analyze the event.

For a locally Lipschitz numerical realization, one may replace a derivative by a directional derivative, Clarke generalized Jacobian, or subdifferential. For a feasible-set valued decision, one may use tangent and normal cones. The infinitesimal decision then becomes set-valued:

\[ \partial U(\theta )[v] \subseteq T U(\theta ). \]

Nonuniqueness records competing local continuations of the decision.

Example 7.4 A switching optimum

Let

\[ V_\theta =\max \{ f_1(\theta ),f_2(\theta )\} . \]

Away from a tie, the tangent follows the active branch. At \(f_1(\theta )=f_2(\theta )\), the generalized derivative retains both branch gradients and their convex combinations. The tie is not numerical noise to be smoothed away automatically: it is a localized change in the candidate cocone and may alter the chosen behavior.

This is the decision-theoretic counterpart of the nonsmooth tangent language introduced in Chapter 1. Smooth infinitesimals and generalized derivatives serve the same structural purpose: they reveal how a declared composite can change locally without pretending that every local response is a single vector.