发表机构
Politecnico di Milano（米兰理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究限价订单簿下在线做市的遗憾最小化，提出基于耦合网格离散化和Hedge的算法，将遗憾界从T^(2/3)改进到√T，并证明完全对抗环境下次线性遗憾不可能。
AI 中文摘要
我们研究做市中的在线学习问题，在每一轮中，做市商在观察市场价格和 incoming 交易者的私有估值之前发布买价和卖价。在此设置下，Maran 等人 2026 引入了一个受限价订单簿启发的反馈模型，其中交易者的估值仅在未发生交易时才会被揭示。假设交易者估值独立同分布地来自一个未知分布，而市场价格由对手方选择，他们建立了期望遗憾界 $\tild{\mathcal{O}}(T^{2/3})$。在这项工作中，我们通过建立高概率遗憾界 $\tild{\mathcal{O}}(\sqrt{T})$ 来改进这一保证。作为热身，我们首先考虑完全反馈设置。我们引入基于两个耦合网格的买卖价差空间的离散化，并将其与 Hedge 结合以实现所需的遗憾率。基于这些想法，我们随后处理由限价订单簿引起的显著较弱的反馈，并开发了一种实现相同保证的算法。最后，我们研究了完全对抗环境中可学习性的极限，其中估值也可能任意变化。令人惊讶的是，我们表明，当市场价格和交易者估值均由对手方选择时，即使在完全反馈下，次线性遗憾也是不可能的，从而激励我们对估值采用随机假设。
英文摘要
We study online learning in market making, where, at each round, a market maker posts bid and ask prices before observing the market price and the private valuation of an incoming trader. In this setting, Maran et al. 2026 introduce a feedback model motivated by limit order books, in which the trader's valuation is revealed only if no transaction occurs. Assuming that trader valuations are drawn i.i.d. from an unknown distribution while market prices are chosen adversarially, they establish an expected regret bound of $\widetilde{\mathcal{O}}(T^{2/3})$. In this work, we improve upon this guarantee by establishing a high-probability regret bound of $\widetilde{\mathcal{O}}(\sqrt{T})$. As a warm-up, we first consider the full-feedback setting. We introduce a discretization of the bid-ask space based on two coupled grids and combine it with Hedge to achieve the desired regret rate. Building on these ideas, we then address the substantially weaker feedback induced by a limit order book and develop an algorithm that achieves the same guarantee. Finally, we investigate the limits of learnability in fully adversarial environments, where the valuations may vary arbitrarily as well. Perhaps surprisingly, we show that when both market prices and trader valuations are chosen adversarially, sublinear regret is impossible even under full feedback, thereby motivating our stochastic assumption on the valuations.
