Learning Shortest Paths When Data is Scarce

TL;DR

This work tackles shortest-path routing when edge costs are observed with bias in a cheap simulator and real measurements are scarce. It models simulator bias on edges as a smooth signal over an edge-similarity graph and estimates it via Laplacian-regularized least squares, yielding calibrated edge costs that improve path decisions. The authors derive finite-sample error bounds, translate them into path-level suboptimality guarantees, and provide a data-driven certificate for near-optimal routes. They also introduce Active Estimated Shortest Path (A-ESP), an active-learning algorithm that adaptively queries edges to identify the true shortest path with high probability in cold-start settings. Empirical results on synthetic road networks and real traffic graphs show improved edge calibration and shortened path costs under data scarcity, with the active learner significantly reducing real-data requirements while converging toward real-data performance as data accumulate.

Abstract

Digital twins and other simulators are increasingly used to support routing decisions in large-scale networks. However, simulator outputs often exhibit systematic bias, while ground-truth measurements are costly and scarce. We study a stochastic shortest-path problem in which a planner has access to abundant synthetic samples, limited real-world observations, and an edge-similarity structure capturing expected behavioral similarity across links. We model the simulator-to-reality discrepancy as an unknown, edge-specific bias that varies smoothly over the similarity graph, and estimate it using Laplacian-regularized least squares. This approach yields calibrated edge cost estimates even in data-scarce regimes. We establish finite-sample error bounds, translate estimation error into path-level suboptimality guarantees, and propose a computable, data-driven certificate that verifies near-optimality of a candidate route. For cold-start settings without initial real data, we develop a bias-aware active learning algorithm that leverages the simulator and adaptively selects edges to measure until a prescribed accuracy is met. Numerical experiments on multiple road networks and traffic graphs further demonstrate the effectiveness of our methods.

Figures (12)

• Figure 1: Topology of the three urban road networks from netzschleuder. Barcelona contains many short loops, Irvine2 is approximately tree-like, and Brasilia combines both patterns.
• Figure 2: Topology of the two real traffic graphs used in our case study. Each edge represents a sensor location that records traffic flow information.
• Figure 3: Effect of the number of real observations on the RMSE of $\hat{\mu}$ for the three netzschleuder graphs (left to right: Barcelona, Brasilia, Irvine2). The horizontal axis is the fraction of observable edges and the number of real samples per observable edge is fixed at $n_e = 20$, thus varying the observable fraction changes the total number of real samples. Results are averaged over multiple seeds (lower is better).
• Figure 4: RMSE of $\hat{\mu}$ across simulator-bias smoothness and magnitude on the three netzschleuder graphs. Each panel sweeps the Laplacian seminorm target $\| b^\star\|_L$ (via $B$) and the smoothing parameter $\rho$ used to generate $b^\star$. Results are averaged over seeds (lower is better).
• Figure 5: Effect of the real-observation noise level on the RMSE of the calibrated edge means $\hat{\mu}$ for the three netzschleuder graphs (left to right: Barcelona, Brasilia, Irvine2). Curves are averaged over multiple random seeds (lower is better).

Theorems & Definitions (17)

• Lemma 1: Uniqueness and closed form
• Theorem 1: High-probability estimation bound
• Theorem 2: Path Suboptimality from Edgewise Errors
• Theorem 3: Anytime pathwise confidence and correctness
• Theorem 4: Sample complexity for A-ESP with global greedy sampling
• Remark 1: Non-unique optimal paths
• proof
• proof
• proof
• proof