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 $L \ge L_{ \mathrm{ ideal}} + g_1(\lambda) + g_2(\varepsilon_\star) + g_{ 12}(\lambda,\varepsilon_\star) - D_{ \mathrm{ ncx}}$, where $g_1$ and $g_2$ are calibrated penalties for latency and order-sensitivity, $g_{ 12}$ captures their geometric interaction, and $D_{ \mathrm{ ncx}}\ge 0$ 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 $2\times 2$ 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.
Categories: cs.LG cs.AI