Algorithmic Robust Forecast Aggregation

TL;DR

This work develops an algorithmic framework for robust forecast aggregation under unknown information structures by casting the problem as a zero‑sum game between nature and an aggregator. It delivers a fully polynomial time approximation scheme (FPTAS) for finite sets of information structures and extends to continuous settings by imposing Lipschitz constraints on aggregators, using dimension reduction, discretization, and online learning to approximate equilibria. In the two‑agent binary‑state model, the framework yields an FPTAS for both discrete reports and Lipschitz aggregators, achieving a near‑optimal regret of about $0.0226$ (very close to the lower bound $\frac{1}{8}(5\sqrt{5}-11) \approx 0.0225$). Numerically, the proposed aggregators outperform simple averaging and prior robust methods, especially in high‑confidence cases, by producing calibrated yet sometimes more extreme forecasts. The approach integrates TVD/EMD covering, coupling arguments, and smoothing of the omniscient posterior to enable scalable robust aggregation with practical implications for weather, epidemiology, and other domains where information structures are uncertain.

Abstract

Forecast aggregation combines the predictions of multiple forecasters to improve accuracy. However, the lack of knowledge about forecasters' information structure hinders optimal aggregation. Given a family of information structures, robust forecast aggregation aims to find the aggregator with minimal worst-case regret compared to the omniscient aggregator. Previous approaches for robust forecast aggregation rely on heuristic observations and parameter tuning. We propose an algorithmic framework for robust forecast aggregation. Our framework provides efficient approximation schemes for general information aggregation with a finite family of possible information structures. In the setting considered by Arieli et al. (2018) where two agents receive independent signals conditioned on a binary state, our framework also provides efficient approximation schemes by imposing Lipschitz conditions on the aggregator or discrete conditions on agents' reports. Numerical experiments demonstrate the effectiveness of our method by providing a nearly optimal aggregator in the setting considered by Arieli et al. (2018).

Figures (12)

• Figure 1: Heatmaps of different aggregators $f(x_1,x_2)$. The horizontal axis represents the first agent's report $x_1$, and the vertical axis represents the second agent's report $x_2$. Darker to lighter shades represent the range of $f(x_1,x_2)$ from 0 to 1.
• Figure 2: Losses of the optimal aggregator under additive, ratio, and absolute robustness paradigms. We pick a finite collection of the information structures, and the horizontal axis represents the information structures sorted by their losses under the omniscient aggregator. The vertical axis represents the loss. The bottom curve (navy blue) represents the optimal loss (lower bound), i.e., the loss of the omniscient aggregator. The middle area (green), is the range of loss of the optimal aggregator $f$ obtained by our algorithm for each paradigm. The top curve (cyan) represents the highest loss that can be afforded for each information structure without increasing the maximum regret. The worst case occurs when the top curve touches the middle curve.
• Figure 3: Sensitivity of the omniscient aggregator $g_\mu({\bm{x}})$ regarding $\mu$. We fix agents' reports ${\bm{x}}=(.2, .2)$ and ${\bm{x}}=(.7, .7)$, and plot the Bayesian posteriors $g_\mu({\bm{x}})$ as function of prior $\mu$. The figures show that 1) surprisingly, as the prior increases, the posterior decreases. Intuitively, fixing the agents' reports, as the prior probability of rain increases, the reports will increasingly resemble negative evidence for rain. Consequently, the true posterior probability should decrease. 2) $g_\mu({\bm{x}})$ is highly sensitive when the prior $\mu$ is near certain and closely aligned with the reports.
• Figure 4: Sensitivity of the omniscient aggregator regarding ${\bm{x}}$: $g_\mu({\bm{x}})$ vs. ${\bm{x}}$. We trim the sensitive area and extend the insensitive area $A_\mu$.
• Figure 5: An example of area division.

Theorems & Definitions (50)

• Definition 3.1: $L$-Lipschitz, sohrab2003basic
• lemma 1: Bayesian Posterior, bordley1982multiplicative
• lemma 2: Reports Distribution, doi:10.1073/pnas.1813934115
• lemma 3: Additive Regret, doi:10.1073/pnas.1813934115
• Definition 4.1: $\epsilon$-Best Response
• Theorem 4.2
• lemma 4
• proof
• Corollary 4.3
• Corollary 4.5