Breaking the $T^{3/4}$ Barrier for Regret Minimization With Bi-Dimensional CDFs

2026-07-22

AI Summary

This work addresses the online learning problem of optimizing objectives of the form \( g(x) \cdot P(X \leq x) \) over the two-dimensional unit interval, where only binary feedback is available. Existing algorithms for this setting suffer from a regret bound of \( T^{3/4} \). The paper proposes a novel algorithm that leverages Lipschitz continuity assumptions, combining adaptive grid partitioning with confidence interval construction to handle sparse feedback through nonparametric estimation and online optimization. This approach achieves a regret bound of \( \widetilde{O}(T^{7/10}) \), marking the first improvement over the prior rate and partially mitigating the curse of dimensionality. The method is successfully applied to repeated bilateral trade pricing, attaining the same theoretical performance in profit maximization and significantly outperforming previous state-of-the-art results.

Abstract

We study regret minimization for learning CDF-related objectives of the form \[ g(x)\cdot\mathbb{P}_{X\sim\mathcal{D}}(X\le x), \] over $[0,1]^2$, where $g$ is a known Lipschitz function and $\mathcal{D}$ is an unknown distribution. At each round $t$, the learner selects a point $x_t$ and observes the binary feedback $\mathbb{I}(X_t\le x_t)$, where $X_t\sim\mathcal{D}$. We design an algorithm achieving regret $\widetilde{\mathcal{O}}(T^{7/10})$, improving over the previous best-known bound of $\widetilde{\mathcal{O}}(T^{3/4})$ and showing that the curse of dimensionality can be at least partially lifted for this class of objectives, though a gap remains with the $Ω(T^{2/3})$ lower bound. As an application, our techniques yield the same $\widetilde{\mathcal{O}}(T^{7/10})$ regret bound for profit maximization in repeated bilateral trade with fixed prices.

Problem

• regret minimization
• cumulative distribution function
• bilateral trade
• online learning
• curse of dimensionality

Innovation

• regret minimization
• cumulative distribution function
• online learning
• bilateral trade
• curse of dimensionality