[ { "answer_declarations": [ { "assumption_deltas": [ { "classification": "generalization", "lean": "C² objective on a complete real Hilbert space.", "reason": "The symmetric Rayleigh quotient norm identity replaces the finite-dimensional eigenvalue description and the same vector segment FTC is used.", "source": "C² Euclidean objective." }, { "classification": "same", "lean": "StrongConvexOn univ α f and the global upper quadratic model in the actual gradient.", "reason": "Under C² regularity, existing compiled C² equivalences give exactly the lower and upper Hessian quadratic-form bounds used by the source.", "source": "α-convex and β-smooth objective." }, { "classification": "source-implicit", "lean": "Actual Riesz Hessian derivative, symmetric continuous linear update derivative, continuous segment derivative.", "reason": "C² and completeness discharge the derivative, symmetry and integrability conditions inside the proof; no extra analytic hypothesis is supplied.", "source": "C² Hessian and segment integral." }, { "classification": "generalization", "lean": "Signed α,β with the actual compatible curvature models and h≥0.", "reason": "The intermediate endpoint argument does not use parameter positivity.", "source": "Positive-curvature condition-number regime." }, { "classification": "same", "lean": "Only the bound component.", "reason": "No claim of objective-specific best step or oracle complexity. The derivative/FTC route is retained. The step-selection obligation belongs to the separate declaration.", "source": "Minimize the bound on the update difference." } ], "audit_id": "ASTIS-RT-ANDI-OPT-GradientEndpoint", "boundary": "The source works in Euclidean space; the Lean theorem extends the same argument to complete real Hilbert spaces using the symmetric Rayleigh norm formula. Completeness supplies the Riesz map and the vector-valued integral. C² regularity supplies the actual gradient derivative, Hessian symmetry, and continuity/integrability along the closed line segment. No additional domination, measurability, boundary or supplied spectral hypothesis is assumed. These are textbook optimisation edges. No minimizer, N-step convergence, stochastic sampling theorem, oracle lower bound or companion-paper completion is asserted. This is the Exercise3.2 derivative/integral proof component before step selection. It does not by itself assert contraction: M_h can exceed one. It includes h=0 and allows signed curvature bounds.", "cell": "ASTIS-SHARED-gradient-step-endpoint-bound", "declaration": "AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentOptimalStep.gradient_step_endpoint_bound", "source_file": "AutoSamplingTheory/TechnicalLemmas/Analysis/GradientDescentOptimalStep.lean", "source_line": 25, "source_sha256_lf": "856f336591e9cd23a69b827d5c438ec70628e530974c4b63981e6b13cd77dc83", "supports": [ "hessian-segment-bound" ] }, { "assumption_deltas": [ { "classification": "generalization", "lean": "C² objective on a complete real Hilbert space.", "reason": "The symmetric Rayleigh quotient norm identity replaces the finite-dimensional eigenvalue description and the same vector segment FTC is used.", "source": "C² Euclidean objective." }, { "classification": "same", "lean": "StrongConvexOn univ α f and the global upper quadratic model in the actual gradient.", "reason": "Under C² regularity, existing compiled C² equivalences give exactly the lower and upper Hessian quadratic-form bounds used by the source.", "source": "α-convex and β-smooth objective." }, { "classification": "source-implicit", "lean": "Actual Riesz Hessian derivative, symmetric continuous linear update derivative, continuous segment derivative.", "reason": "C² and completeness discharge the derivative, symmetry and integrability conditions inside the proof; no extra analytic hypothesis is supplied.", "source": "C² Hessian and segment integral." }, { "classification": "generalization", "lean": "0≤α≤β and β>0; optimal comparison for all real h.", "reason": "At α=0 the extension is nonexpansive and no κ is formed. β>0 gives a positive denominator; α≤β is explicit rather than inferred in the zero-dimensional space. For α>0 this is the source contraction factor.", "source": "Positive-curvature condition-number regime." }, { "classification": "same", "lean": "Pairwise bound at 2/(α+β) and global scalar minimax inequality for the endpoint envelope.", "reason": "No claim of objective-specific best step or oracle complexity. The derivative/FTC route is retained. The scalar lower bound is proved for all real h, so in particular covers nonnegative descent steps.", "source": "Minimize the bound on the update difference." } ], "audit_id": "ASTIS-RT-ANDI-OPT-GradientOptimalStep", "boundary": "The source works in Euclidean space; the Lean theorem extends the same argument to complete real Hilbert spaces using the symmetric Rayleigh norm formula. Completeness supplies the Riesz map and the vector-valued integral. C² regularity supplies the actual gradient derivative, Hessian symmetry, and continuity/integrability along the closed line segment. No additional domination, measurability, boundary or supplied spectral hypothesis is assumed. These are textbook optimisation edges. No minimizer, N-step convergence, stochastic sampling theorem, oracle lower bound or companion-paper completion is asserted. For the source’s condition-number interpretation assume α>0: κ=β/α and q=(κ−1)/(κ+1)<1. At α=0 the extension gives q=1 (nonexpansive) and κ is undefined. At α=β>0 the map has zero pairwise difference; this does not identify a minimizer without further argument. The explicit order α≤β preserves the zero-dimensional case, where curvature inequalities alone do not imply this order. The result minimizes the endpoint upper envelope; it does not state that every individual objective has its actual smallest Lipschitz constant at h★.", "cell": "ASTIS-SHARED-gradient-descent-optimal-step", "declaration": "AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentOptimalStep.optimal_gradient_step", "source_file": "AutoSamplingTheory/TechnicalLemmas/Analysis/GradientDescentOptimalStep.lean", "source_line": 102, "source_sha256_lf": "856f336591e9cd23a69b827d5c438ec70628e530974c4b63981e6b13cd77dc83", "supports": [ "balanced-minimax-step" ] } ], "book": { "authors": [ "Sinho Chewi" ], "edition": "arXiv:2605.07006v1", "html_url": "https://arxiv.org/html/2605.07006v1#S3", "pdf_pages": 149, "pdf_sha256": "9e28afeacb0121b79691fadc6af9c145ccd787bffd2cf9c3f2310806112e644b", "pdf_url": "https://arxiv.org/pdf/2605.07006v1", "rights_boundary": "The source has an arXiv non-exclusive distribution grant. No blanket source-text/solution-manual ML reuse license is inferred; only original mathematical restatement, original solutions and bibliographic correspondence enter the Bench. The PDF/render stays outside ML exports.", "title": "Lectures on Optimization" }, "book_id": "chewi-optimization-v1", "book_page": 23, "canonical_source": { "anchor": "Exercise3.2 and the Section3 C² convention; Definition1.12 smoothness", "attribution": "Sinho Chewi, Lectures on Optimization. ASTIS-authored paraphrase; no endorsement implied.", "edition": "arXiv:2605.07006v1", "url": "https://arxiv.org/html/2605.07006v1#S3", "wording_status": "faithful paraphrase" }, "chapter": "3", "context_notes": [ "Chapter 3 explicitly assumes C² regularity. The condition-number interpretation uses 0<α≤β and κ=β/α.", "The owned APIs generalize Euclidean space to complete real Hilbert spaces, and retain signed-modulus or α=0 intermediate extensions. Those are disclosed API generalizations, not changes to the source exercise.", "The answer minimizes a uniform curvature-endpoint bound, not the actual optimal Lipschitz constant of each objective." ], "formal_scope": "named owned core answers; not full exercise/formal QA certification", "id": "chewi-opt-v1-exercise-3-2", "number": "3.2", "owned_answer_build_recorded": false, "parts": [ { "id": "hessian-segment-bound", "label": "Derive the endpoint Lipschitz bound from the actual Hessian and segment integral" }, { "id": "balanced-minimax-step", "label": "Select the balanced step and prove minimality of its curvature envelope" } ], "pdf_page": 23, "publication_file": "website/content/publications/optimisation.json", "questions": [ { "core_id": "sogb-029", "editorial": "accepted-after-repair", "tracks": { "natural-language": "L3" } } ], "source_assumptions": [ "The surrounding section works with C² Euclidean objectives, an α-strong-convexity lower bound and β-smoothness upper bound.", "The condition-number contraction formula is used in the positive-curvature regime 0<α≤β; κ=β/α. Step h is a nonnegative gradient-descent step in the derivative-bound argument." ], "source_entry_sha256": "17c775ef4f10f62057023351cf9aa126d45ebb59f67fc3eed06eee4cdfbb6455", "source_statement": "For an α-convex and β-smooth C² Euclidean objective, apply the fundamental theorem of calculus to T_h=id−h∇f along the segment between two points. Bound the eigenvalues of DT_h to obtain its operator norm bound and show that h=2/(α+β) minimizes the resulting bound. The resulting factor is (κ−1)/(κ+1), where κ=β/α, improving the preceding gradient-descent estimate.", "unformalized_context": [ "The source's informal large-condition-number iteration-complexity comparison is contextual commentary, not an output of the two named owned declarations.", "The finite-dimensional eigenvalue presentation and κ notation are explained in the mathematical answer; the actual owned proofs use the symmetric Rayleigh norm formula and the equivalent (β−α)/(α+β) expression. No separate eigenvalue-list or asymptotic-complexity declaration is claimed." ], "wording_status": "original faithful mathematical restatement; not reproduced source prose" } ]