Latency and Ordering Effects in Online Decisions

TL;DR

This paper develops a theoretical framework for analyzing the impact of latency and order effects in online decision systems, providing bounds and practical diagnostics for real-world applications.

Contribution

It introduces a structured lower bound on excess loss incorporating latency and order-sensitivity, extending to nonconvex settings and offering practical estimation methods.

Findings

Structured lower bound on excess loss with latency and order penalties

Extension of bounds to nonconvex and weakly convex settings

Operational diagnostics for estimating and monitoring effects

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 $D_\Phi$ 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.