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Bilateral Trade Under Heavy-Tailed Valuations: Minimax Regret with Infinite Variance

Hangyi Zhao

TL;DR

The self-bounding property of Bachoc et al. (ICML 2025) is extended from bounded to real-valued valuations, showing that the expected regret of any price $\pi$ satisfies $\mathbb{E}[g(m,V,W) - g(\pi,V,W) - g(\pi,V,W)] under bounded density alone.

Abstract

We study contextual bilateral trade under full feedback when trader valuations have bounded density but infinite variance. We first extend the self-bounding property of Bachoc et al. (ICML 2025) from bounded to real-valued valuations, showing that the expected regret of any price $π$ satisfies $\mathbb{E}[g(m,V,W) - g(π,V,W)] \le L|m-π|^2$ under bounded density alone. Combining this with truncated-mean estimation, we prove that an epoch-based algorithm achieves regret $\widetilde{O}(T^{1-2β(p-1)/(βp + d(p-1))})$ when the noise has finite $p$-th moment for $p \in (1,2)$ and the market value function is $β$-Hölder, and we establish a matching $Ω(\cdot)$ lower bound via Assouad's method with a smoothed moment-matching construction. Our results characterize the exact minimax rate for this problem, interpolating between the classical nonparametric rate at $p=2$ and the trivial linear rate as $p \to 1^+$.

Bilateral Trade Under Heavy-Tailed Valuations: Minimax Regret with Infinite Variance

TL;DR

The self-bounding property of Bachoc et al. (ICML 2025) is extended from bounded to real-valued valuations, showing that the expected regret of any price satisfies $\mathbb{E}[g(m,V,W) - g(\pi,V,W) - g(\pi,V,W)] under bounded density alone.

Abstract

We study contextual bilateral trade under full feedback when trader valuations have bounded density but infinite variance. We first extend the self-bounding property of Bachoc et al. (ICML 2025) from bounded to real-valued valuations, showing that the expected regret of any price satisfies under bounded density alone. Combining this with truncated-mean estimation, we prove that an epoch-based algorithm achieves regret when the noise has finite -th moment for and the market value function is -Hölder, and we establish a matching lower bound via Assouad's method with a smoothed moment-matching construction. Our results characterize the exact minimax rate for this problem, interpolating between the classical nonparametric rate at and the trivial linear rate as .
Paper Structure (9 sections, 4 theorems, 38 equations)

This paper contains 9 sections, 4 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Related work
  3. Setup and the Structural-Algorithmic Gap
  4. Main Results
  5. Proof of Theorem \ref{['thm:parametric']}
  6. Proof of Theorem \ref{['thm:nonparametric']}
  7. Lower Bound
  8. Discussion
  9. Proof of Lemma \ref{['lem:selfbound']}

Key Result

Lemma 3.1

Under Assumption ass:density with $\mathop{\mathrm{\mathbb{E}}}\limits[|\xi_t|],\mathop{\mathrm{\mathbb{E}}}\limits[|\zeta_t|] < \infty$ (implied by $p > 1$), for all $\pi \in \mathbb{R}$: Moreover, $m$ uniquely maximizes $\pi \mapsto \mathop{\mathrm{\mathbb{E}}}\limits[g(\pi,V,W)]$, and $\mathop{\mathrm{\mathbb{E}}}\limits[g(m,V,W)] \le 2\sigma_p$.

Theorems & Definitions (9)

  • Remark 2.4: Two-bit feedback
  • Lemma 3.1: Generalized self-bounding property
  • proof : Proof sketch (full proof in Appendix \ref{['app:selfbound']})
  • Theorem 3.2: Parametric
  • Theorem 3.3: Nonparametric
  • Remark 3.4: Rate comparison
  • Proposition 6.1
  • proof
  • Remark 6.2: Parametric lower bound