{"ID":6023427,"CreatedAt":"2026-07-08T01:00:23.257252134Z","UpdatedAt":"2026-07-10T07:42:22.599444549Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2607.05947","arxiv_id":"2607.05947","title":"Higher-Order Derivatives Do Not Accelerate the Computation of Fixed Points","abstract":"The Picard iteration converges to the unique fixed point of a $q$-contractive operator at a linear rate $q^N$, and a lower bound with an affine construction shows that no deterministic method querying only operator values can do better. But what about higher-order methods that query derivatives? A single Jacobian evaluation reveals an affine map entirely, so the affine construction says nothing about higher-order methods. In this work, we show that finite-order derivative information still does not accelerate the worst-case complexity for smooth contractive fixed-point computation. This contrasts with higher-order smooth minimization, where higher-order derivatives do improve worst-case rates for convex and non-convex minimization.","short_abstract":"The Picard iteration converges to the unique fixed point of a $q$-contractive operator at a linear rate $q^N$, and a lower bound with an affine construction shows that no deterministic method querying only operator values can do better. But what about higher-order methods that query derivatives? A single Jacobian evalu...","url_abs":"https://arxiv.org/abs/2607.05947","url_pdf":"https://arxiv.org/pdf/2607.05947v1","authors":"[\"Uijeong Jang\",\"Ernest K. Ryu\"]","published":"2026-07-07T07:47:13Z","proceeding":"math.OC","tasks":"[\"math.OC\"]","methods":"[]","has_code":false}