Under the assumption that the out-of-distribution-generalizing model is the simplest source-consistent parameter, regularized maximum likelihood recovers it with near-1/n excess risk (constant gap) or n^{-1+2/(3τ)} (vanishing gap).
Since f is smooth, the first-order optimality condition implies that its directional derivative vanishes along directions in the tangent space: 2⟨β − β′, v⟩ = 0, ∀v ∈ T(β′)
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Principled Out-of-Distribution Generalization via Simplicity
Under the assumption that the out-of-distribution-generalizing model is the simplest source-consistent parameter, regularized maximum likelihood recovers it with near-1/n excess risk (constant gap) or n^{-1+2/(3τ)} (vanishing gap).