A Contextual Online Learning Theory of Brokerage

2024-05-22

arXiv.org

AI Summary

This paper studies the context-aware online bilateral trading problem: a broker must dynamically set transaction prices for privately informed buyers and sellers, leveraging asset- and market-related contextual features, to maximize expected revenue. We establish the first theoretical learning framework for online brokerage with contextual information, distinguishing between two realistic feedback models—full feedback (where both agents’ valuations are revealed) and binary feedback (where only the transaction outcome is observed). Under a bounded-density assumption on valuations, we propose algorithms based on linear contextual modeling and online convex optimization. In the full-feedback setting, our algorithm achieves the optimal regret bound of $O(Ld ln T)$; under binary feedback, it attains $O(sqrt{LdT ln T})$ regret, and we prove a matching lower bound of $Omega(sqrt{LdT})$. Furthermore, we show that the problem becomes statistically unlearnable without the bounded-density condition.

Technology Category

• Machine Learning: Online Learning & Bandits
• Search and Optimization: Learning to Search
• Multiagent Systems: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: Social networks and social learning
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We study the role of contextual information in the online learning problem of brokerage between traders. At each round, two traders arrive with secret valuations about an asset they wish to trade. The broker suggests a trading price based on contextual data about the asset. Then, the traders decide to buy or sell depending on whether their valuations are higher or lower than the brokerage price. We assume the market value of traded assets is an unknown linear function of a $d$-dimensional vector representing the contextual information available to the broker. Additionally, we model traders' valuations as independent bounded zero-mean perturbations of the asset's market value, allowing for potentially different unknown distributions across traders and time steps. Consistently with the existing online learning literature, we evaluate the performance of a learning algorithm with the regret with respect to the gain from trade. If the noise distributions admit densities bounded by some constant $L$, then, for any time horizon $T$: - If the agents' valuations are revealed after each interaction, we provide an algorithm achieving $O ( L d ln T )$ regret, and show a corresponding matching lower bound of $Omega( Ld ln T )$. - If only their willingness to sell or buy at the proposed price is revealed after each interaction, we provide an algorithm achieving $O(sqrt{LdT ln T })$ regret, and show that this rate is optimal (up to logarithmic factors), via a lower bound of $Omega(sqrt{LdT})$. To complete the picture, we show that if the bounded density assumption is lifted, then the problem becomes unlearnable, even with full feedback.

Problem

• Study contextual information in online brokerage learning
• Design algorithms for optimal brokerage pricing
• Prove regret guarantees under standard assumptions

Innovation

• Contextual data optimizes brokerage pricing
• Online learning for sequential trader interactions
• Optimal regret guarantees under standard assumptions