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Fenchel-Young Duality Gaps: Certified Early Stopping for Regularized Inverse Problems

arXiv · AI, language, vision and robotics · article · Sep 15, 2026 · UTC

We study computable error bounds and certified early stopping for regularized inverse problems, where a data-fidelity term is traded against a regularizer. The analysis relies on an exact duality-gap identity that splits the total gap of $F(Φμ)+λR(μ)$ into a data-fidelity Fenchel--Young loss and a regularizer Fenchel--Young loss, $ Δ(μ,h)=L_F(Φμ\parallel h)+λL_R(μ\parallelη),\qquad η=-Φ^\star h/λ, $ valid for any primal point $μ$ and any dual point $h$. The data-fidelity term $F$ is strictly convex, so wherever $F^\star$ is differentiable the loss $L_F(Φμ\parallel h)$ is the Bregman divergence

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