Experimentation Under Non-stationary Interference

TL;DR

This work addresses estimation of the average treatment effect in multi-period experiments with non-stationary spatio-temporal interference, where interference graphs evolve independently of treatment. It introduces a truncated Horvitz-Thompson estimator and a Last Interaction Time (LIT) based covariance bound, proving that the MSE decays linearly with the number of space-time blocks and is scaled by a graph-structure factor captured by the average cluster degree. The analysis leverages clustering-induced graphs (CIGs) under vertical designs, yielding a variance bound that scales with $(NT)^{-1}$ and the average cluster degree, and a bias bound that decays exponentially with the truncation radius $r$ relative to the mixing time $t_{\rm mix}$. The results apply to concrete settings such as metric-space interference and dynamic Erdos-Renyi graphs, providing practical guidelines for constructing exposure mappings and obtaining finite-sample guarantees in dynamic online environments.

Abstract

We study the estimation of the ATE in randomized controlled trials under a dynamically evolving interference structure. This setting arises in applications such as ride-sharing, where drivers move over time, and social networks, where connections continuously form and dissolve. In particular, we focus on scenarios where outcomes exhibit spatio-temporal interference driven by a sequence of random interference graphs that evolve independently of the treatment assignment. Loosely, our main result states that a truncated Horvitz-Thompson estimator achieves an MSE that vanishes linearly in the number of spatial and time blocks, times a factor that measures the average complexity of the interference graphs. As a key technical contribution that contrasts the static setting we present a fine-grained covariance bound for each pair of space-time points that decays exponentially with the time elapsed since their last ``interaction''. Our results can be applied to many concrete settings and lead to simplified bounds, including where the interference graphs (i) are induced by moving points in a metric space, or (ii) follow a dynamic Erdos-Renyi model, where each edge is created or removed independently in each time period.

Figures (3)

• Figure 1: Naive Covariance Bound Is Loose. Wlog assume that $t'\le t$. The naive covariance bound only gives $e^{-|t-t'|/t_{\rm mix}}$ (see \ref{['eqn:111324']}), which can be loose if $t\approx t'$. To tighten this, we will define the last interaction time $\uptau^\star(i,i';t')$ and show that the covariance decays exponentially in $t -\uptau^\star(i,i';t')$.
• Figure 2: Clustering-induced Graphs. To visualize, we spread out the individuals along a one-dimensional line (the vertical axis). Observe from that at time $\tau\in [(k-1)\ell, k\ell]$, there is an edge between $i$ and some individual $j$ in the spatio-block $C\in \Pi_k$, and similarly, $i'$ has an edge with some $j'\in C$ at $\tau'\in [(k-1)\ell, k\ell]$. Therefore, we include $(i,i')$ as an edge in the $k$-th CIG.
• Figure 3: CIG in a Metric Space. To visualize, let us flatten the metric space onto a 1-d line (the vertical axis). In this figure $(i,i')$ is an edge in the $k$-th CIG. In fact, at time $\tau\in [(k-1)\ell, k\ell]$, individual $i$ is within distance $\kappa$ to another individual $j$, who was in spatio-block $C$ at the starting time $(k-1)\ell$ of the time block. Similarly, $i'$ is within distance $\kappa$ to $j'$, which is also inside $C$ at time $(k-1)\ell$. Thus, both $i,i'$ depend on the treatment assigned to $C$ throughout the entire time block, leading to a correlation between their future outcomes and exposure mappings.

Theorems & Definitions (28)

• Definition 2.2: Average Treatment Effect
• Proposition 2.3: Initial State Doesn't Matter Much
• Definition 3.1: Vertical Design
• Definition 3.2: Spatio-temporal Neighborhood
• Definition 3.3: Exposure Mapping
• Definition 3.4: Horvitz-Thompson Estimator
• Proposition 3.5: Bias bound
• Definition 3.6: Clustering-Induced Graph
• Definition 3.7: Region-based Vertical Design
• Definition 4.1: Last Interaction Time