Parameter-Efficient Is Not Function-Stable: Function Trust for Model Adaptation

Parameter efficiency constrains where a model changes, but not how far its predictive function moves. We introduce \trust, a parameterization-agnostic framework that adapts a model subject to a user-specified budget on the forward KL divergence from a frozen source predictor over cached source inputs. A primal–dual Powell–Hestenes–Rockafellar augmented Lagrangian approximately solves the constrained problem, while the budget formulation remains independent of the solver. Conditional on achieving the replay constraint within tolerance $\xi$, we bound population functional drift by ${\rho }_{\mathrm{KL}}+\xi +{\Gamma }_n$ and source-risk change by $L\sqrt{({\rho }_{\mathrm{KL}}+\xi +{\Gamma }_n)/2}$, where ${\Gamma }_n$ captures replay-to-population generalization; a margin corollary controls deterministic prediction changes. Across TinyBERT and Qwen3-0.6B, \trust consistently reduces source forgetting and held-out KL for LoRA, full fine-tuning, and our structured adapter, with small or parameterization-dependent changes in target quality. Yet equal replay budgets need not imply equal held-out behavior: increasing TinyBERT replay from $64$ to $4096$ examples reduces validation KL from $0.0544$ to $0.0255$, while low-margin replay at fixed $n=256$ reduces it from $0.0421$ to $0.0209$. Fixed-penalty and augmented-Lagrangian frontiers overlap; our contribution is therefore not solver dominance. The evidence instead shows that functional stability is not a by-product of parameter efficiency: it must be explicitly budgeted, measured, and audited.