8.6 Natural DLINCS updates

Chapter 4 introduced Natural LINCS as metric-aware typed repair. For a stochastic deep model with parameter object \(P\) and conditional law \(p_\theta (y\mid x)\), a canonical candidate geometry is the Fisher metric

where \(u,v\in T_\theta P\), and where the sampling law \(\mu _\theta \) is part of the declaration. For a scalar objective, the coordinate form of the resulting update is

More generally, Deep LINCS substitutes the Fisher metric into the constrained or regularized typed repair problems of Equations 4.1 and 4.2. The structural obstruction need not be collapsed into a single loss before the repair direction is selected.

The pullback Fisher metric of an overparametrized network is commonly singular: two parameter directions may induce the same observable model. From the LINCS perspective this is a quotient, not merely a numerical defect. The compiler should identify parameter gauges and observational null directions, solve on the quotient when possible, and record any practical approximation—pseudoinverse, damping, or structured curvature estimate—in the admission certificate.

This specialization also connects directly to reinforcement learning. On a policy manifold, the Fisher metric is weighted by a declared state-visitation law. Natural policy-gradient and natural actor–critic methods use that geometry to choose an intrinsic actor update. In a Natural-GIRL realization, the Bellman or intervention sketch still declares the repair target; policy geometry selects the actor direction, while the critic and compatible features provide an estimator. Geometry does not confer a causal meaning on the update or certify its admission.