The paper introduces computable error bounds and a certified early‑stopping criterion for regularized inverse problems by exploiting an exact Fenchel–Young duality‑gap identity. The total duality gap splits into a data‑fidelity loss and a regularizer loss, both expressed as Fenchel–Young losses that are oracle‑free and vanish exactly at Mirror Alignment. Using a constructive Brønsted–Rockafellar approach, the authors build a dual‑feasible proxy via a proximal step in the fidelity geometry, enabling an early‑stopping rule based on the regularizer loss.
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