The Sample Complexity of Uniform Approximation for Multi-Dimensional CDFs and Fixed-Price Mechanisms

We study the sample complexity of learning a uniform approximation of an $n$-dimensional cumulative distribution function (CDF) within an error $ε> 0$, when observations are restricted to a minimal one-bit feedback. This serves as a counterpart to the multivariate DKW inequality under ‘‘full feedback’’, extending it to the setting of ‘‘bandit feedback’’. Our main result shows a near-dimensional-invariance in the sample complexity: we get a uniform $ε$-approximation with a sample complexity $\frac{1}{ε^3}{\log\left(\frac 1 ε\right)^{\mathcal{O}(n)}}$ over a arbitrary fine grid, where the dimensionality $n$ only affects logarithmic terms. As direct corollaries, we provide tight sample complexity bounds and novel regret guarantees for learning fixed-price mechanisms in small markets, such as bilateral trade settings.

💡 Research Summary

The paper investigates the sample complexity of uniformly approximating an n‑dimensional cumulative distribution function (CDF) when the learner receives only a single bit of feedback per query—a setting the authors refer to as “bandit feedback.” In the classic full‑feedback scenario, the multivariate Dvoretzky‑Kiefer‑Wolfowitz (DKW) inequality guarantees that O((1/ε²)·log(n/δ)) i.i.d. samples suffice to achieve sup‑norm error ε with probability 1–δ. By contrast, the bandit model only reveals the indicator I