Predictive entropy search for multi-objective Bayesian optimization

⚠️ Curated Limitations & Negative Results

• The ideal target assumes a well-defined joint posterior over objectives and Pareto sets. A bounded domain alone does not establish existence or regularity. Tied objective vectors, repeated input locations and degenerate covariance matrices require explicit handling; no general tie policy or empty-set fallback is invented here.
• Main PDFp.1 gives a correct weak-plus-one-strict dominance definition, but its subsequent strict quantified characterization includes self comparisons. Main PDFp.3 Eq4 includes self pairs with Theta(0)=1 and reverses the minimization comparison direction; Eq6 on p.4 and supplement Eq8 on p.3 repeat that orientation. Supplement Eq3 on p.2 excludes self pairs. Even correctly oriented nondominated-membership factors do not require a supplied subset to equal the entire Pareto set. These printed products are not certified normalized densities or exact set-equality events.
• The supplement has additional literal Gaussian-parameter inconsistencies: PDFp.1 Eq2 cross-term scaling differs from the symmetric precision entries in p.2 Eq4; Eq4 includes an M+M endpoint where the observed block has N+M variables; p.3 Eq7 mixes covariance and precision entries and does not in general construct the precision of the required marginal. The high-level Gaussian projection is curated without certifying these expressions as executable formulas.
• Selective static inspection of the separately pinned July2016 author implementation supports minimization sign reversal, self-site removal and covariance block inversion. That version also rescales candidate covariances, uses a damped single candidate pass and clips each sampled contribution below zero before averaging. These are version-specific implementation choices, not an official erratum or proof of equivalence to the proceedings equations. Conditioning reduces entropy in expectation, so per-realization clipping is an additional approximation. No author code was executed.
• Finite random features, approximate multi-objective optimization, finite Pareto representatives, EP factorization, limited candidate updates and hyperparameter averaging all introduce approximation. The original bounded checks in research/discovery/pesmo-boundaries.py illustrate selected source and information boundaries only; they do not validate full PESMO inference or its empirical accuracy.
• The paper gives linear dependence on the number of objectives only for specified EP/variance linear-algebra operations with other sizes and iteration/sample budgets held fixed. Pareto-set size, input dimension, posterior sampling and acquisition maximization can dominate cost. No universal runtime improvement, convergence guarantee or benchmark reproduction is asserted.