A Contextual Online Learning Theory of Brokerage
Keywords: contextual bandits, bilateral trade, regret minimization, theory
TL;DR: We study a contextual version of online learning for brokerage
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, at each time step, we model traders' valuations as independent bounded zero-mean perturbations of the asset's current 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.
Primary Area: learning theory
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Submission Number: 7068
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