ora-0171

15.4 The weak-learning interface

Let \(\mathcal A\) be an adversary or residual category. A weak learner is not merely a function returning \(h\in H\); it is an open learner whose request map accepts a residual or reweighted presentation and whose response includes a generator together with a progress witness. The booster composes these interfaces and stores a formal distribution \(\mu _N\in D(H)\). Evaluation occurs only through \(\overline i(\mu _N)\), followed, when necessary, by a feasibility repair \(\pi \).

Definition 15.5 Amplification certificate

An amplification certificate for stages \(1,\ldots ,N\) consists of a potential \(\Phi _n\geq 0\), weak progress witnesses \(w_n\), a composition law producing \(\mu _n\in D(H)\), and a repair comparison \(\pi \overline i(\mu _n)\to \overline i(\mu _n)\) such that

\[ \Phi _{n+1}\leq (1-\kappa \gamma _n^2)\Phi _n+\varepsilon _n, \]

where \(\gamma _n\) is the certified weak advantage, \(\kappa {\gt}0\) is uniform, and \(\varepsilon _n\) bounds the defect introduced by approximation and feasibility repair.

Theorem 15.6 Conditional weak-to-strong amplification

If \(\gamma _n\geq \gamma {\gt}0\), \(0{\lt}\kappa \gamma ^2{\lt}1\), and the amplification certificate holds, then

\[ \Phi _N \leq e^{-\kappa \gamma ^2N}\Phi _0 +\sum _{n=0}^{N-1}e^{-\kappa \gamma ^2(N-1-n)}\varepsilon _n. \]

In particular, exact compatible repair \((\varepsilon _n=0)\) yields exponential amplification of the certified potential.

Proof ▼

Iterate the one-step inequality and use \(1-\kappa \gamma ^2\leq e^{-\kappa \gamma ^2}\). The convolution term records the accumulated repair defects instead of silently absorbing them into a scalar loss bound.

The theorem is deliberately conditional: category theory supplies the free composite and the typing of its witnesses, but the weak advantage and potential contraction remain analytic facts. A complete categorical online boosting theorem must derive the certificate from an explicit weak-learning interface and prove that the loss extension and projection contribute the claimed \(\varepsilon _n\).