VABO: Violation-Aware Bayesian Optimization for Closed-Loop Control Performance Optimization with Unmodeled Constraints

Proposition 1 in the paper claims a per-horizon guarantee but its proof silently replaces the per-iteration product-of-marginals constraint (7b) with a lower bound on the joint event without justification. The algorithm’s feasibility set Θ_t imposes ∏_i P( c̄_i(θ_t) ≤ β_{i,t}B_{i,t} | history ) ≥ 1 − ε_t (see 7b and Algorithm 1), not P(⋂_i {c̄_i(θ_t) ≤ β_{i,t}B_{i,t}} | history) ≥ 1 − ε_t. The short proof of Proposition 1 directly uses the latter form, leaving a gap. The candidate model fills this gap correctly using a union bound plus AM ≥ GM and Bernoulli’s inequality to convert the product constraint into a valid joint probability bound at each iteration, after which the tower property yields the stated guarantee. Hence, the proposition is correct, but the paper’s proof is incomplete; the model’s solution supplies the missing step.