Latency and Ordering Effects in Online Decisions

TL;DR

This work develops a structured, geometry-based decomposition of the excess loss in online decision systems subjected to latency and order (noncommutativity) effects. Building on Bregman divergences and two-stage projections, it proves that the regret relative to an ideal Bayes benchmark can be bounded below by additive penalties for latency $g_1(\lambda)$, order-sensitivity $g_2(\varepsilon_*)$, and their interaction $g_{12}(\lambda,\varepsilon_*)$, with a nonconvexity penalty $\Delta_{ncx}$ in nonconvex settings. The authors provide practical pathways to estimate these terms via a 2×2 experimental design, robust reporting through $\mathrm{LB}_{\mathrm{safe}}$, and diagnostic tools (ESS, clipping, heatmaps), along with extensions to prox-regular and geodesic-convex (mirror) settings. The framework unifies latency, noncommutativity, and implementation gaps into a single lower-bound statement and offers guidance matrices for deploying these diagnostics in real systems, enabling principled tuning and risk-controlled experimentation. Overall, the paper delivers a theoretically principled, operationally actionable calculus for diagnosing and mitigating time- and order-induced losses in online decision pipelines.

Abstract

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 $D_Φ$ as the loss benchmark, we prove that the excess benchmark loss admits a structured lower bound $L \ge L_{\mathrm{ideal}} + g_1(λ) + g_2(\varepsilon_\star) + g_{12}(λ,\varepsilon_\star) - D_{\mathrm{ncx}}$, 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.

Figures (1)

• Figure 1: Gaussian toy example under log-loss: the latency penalty $g_1(\lambda)$ (baseline) and the total structural penalty $g_1 + g_2 + g_{12}$ for two levels of geometric/order constraint $\varepsilon$. Both $g_1$ and the total penalty are monotone in the proxy noise scale $\lambda$, and the gap between the curves grows with the strength of the geometric constraint. Reproducible code is provided in Appendix \ref{['app:gaussian-toy-script']}.

Theorems & Definitions (36)

• Definition 1: Interaction term
• Lemma 1: When the interaction vanishes
• Definition 2: Ideal benchmark
• Remark 1: On the interpretation of L_ideal
• Definition 3: Latency-feasible set $\mathcal{C}_{\lambda}$
• Definition 4: Order-feasible set $\mathcal{C}_{\varepsilon}$
• Lemma 2: Generalized Pythagorean inequality
• proof
• Remark 2: Safe reporting
• Theorem 1: Structured Lower-Bound Decomposition