Latency and Ordering Effects in Online Decisions

Online decision systems routinely operate under delayed feedback and order-sensitive (noncommutative) dynamics: actions affect which observations arrive, and in what sequence. Taking a Bregman divergence D_Phi as the loss benchmark, we prove that the excess benchmark loss admits a structured lower bound L ≥ Lideal + g1(λ) + g2(varepsilon_star) + g12(λ,varepsilon_star) - Dncx, where g1 and g2 are calibrated penalties for latency and order-sensitivity, g12 captures their geometric interaction, and Dncx≥ 0 is a nonconvexity/approximation penalty that vanishes under convex Legendre assumptions. We extend this inequality to prox-regular and weakly convex settings, obtaining robust guarantees beyond the convex case. We also give an operational recipe for estimating and monitoring the four terms via simple 2× 2 randomized experiments and streaming diagnostics (effective sample size, clipping rate, interaction heatmaps). The framework packages heterogeneous latency, noncommutativity, and implementation-gap effects into a single interpretable lower-bound statement that can be stress-tested and tuned in real-world systems.