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

TL;DR

This paper analyzes the number of samples needed to approximate multi-dimensional CDFs with minimal feedback, revealing near-dimension-invariant complexity and applying results to fixed-price mechanism learning.

Contribution

It extends the multivariate DKW inequality to bandit feedback, establishing near-dimension-invariant sample complexity bounds and deriving new guarantees for fixed-price mechanisms.

Findings

Sample complexity is nearly independent of dimension, depending mainly on psilon^3 and logarithmic factors.

Provides tight bounds for learning CDFs with one-bit feedback in high dimensions.

Applies results to derive regret guarantees for fixed-price mechanisms in small markets.

Abstract

We study the sample complexity of learning a uniform approximation of an $n$-dimensional cumulative distribution function (CDF) within an error $\epsilon > 0$, 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 $\epsilon$-approximation with a sample complexity $\frac{1}{\epsilon^3}{\log\left(\frac 1 \epsilon \right)^{\mathcal{O}(n)}}$ 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.