Information-geometric optimization

⚠️ Curated Limitations & Negative Results

• The source is the official 2017 JMLR article. Supporting research read native text on pages 1–33, 39–45 and 52–55 and selected visuals; root reading has its own narrower ledger. This is not a full 65-page proof audit. Earlier versions, author implementations and benchmark results were not reproduced.
• The core score/Fisher formulas require a suitable regular sampling family and legitimate differentiation under the integral. An empirical Fisher estimate can be singular: at least as many distinct samples as parameter dimensions is necessary, not sufficient, for invertibility. Arbitrarily large finite steps can leave the parameter domain or poorly approximate the flow; no automatic regularization or boundary repair is supplied.
• The rank utility depends on the current distribution and the selected weight function. A constant weight provides no population preference signal, although its finite-sample estimate can fluctuate. Finite ties average weights over compatible ranks. The integral-weight alternative in footnote 2 is not silently substituted for midpoint weights; they need not agree at finite sample size.
• Definition 3 on page 9 uses a non-increasing weight for minimization. Proposition 27 on page 52 prints non-decreasing, while its proof on pages 53–54 uses non-increasing inequalities. The record follows the defining minimization convention and retains this source inconsistency; it does not claim an author-issued erratum or certification of all proofs.
• The introductory discussion on page 8 uses minimize/descent wording beside a plus-gradient update. Definitions 3–5 specify ascent of the transformed rank utility. The overview on page 17 also needs the detailed finite-step invariance qualifications of Propositions 18–19 on pages 43–44; nonlinear coordinate changes are not asserted to leave arbitrary finite updates identical.
• Strictly increasing objective transformations preserve order and ties; a merely non-decreasing transformation can create new ties. Search-space transformations must preserve the distribution family. Sampling consistency is a fixed-parameter limit and supplies no explicit finite population threshold, high-probability trajectory bound, finite-sample monotonicity or global convergence guarantee.
• IGO-ML is a separately defined likelihood-maximization construction, not an automatic alias of the generic sampled Euler rule. Its required maximizer must be well defined and tractable. Expectation coordinates, mean/covariance coordinates and other smoothed cross-entropy parametrizations need not produce the same finite update. No distinct IGO-ML, rank-mu CMA or PBIL variant record is created by this framework description.
• Existing NES, CMA-ES, cross-entropy optimization and PBIL records remain distinct. The special-case correspondences do not import complete evolution-path adaptation, probability-vector mutation, all raw-fitness NES rules or every cross-entropy variant. No historical-priority, general superiority, diversity-optimality or benchmark claim is made.