---
title: "Blackwell可逼近性与梯度均衡的等价性研究"
source: "arXiv AI/CL/LG（论文）"
source_tier: "t2"
url: "https://arxiv.org/abs/2606.27315v1"
permalink: "https://feed.astralor.com/items/itm_a460bc7beaee6fd7867f6851"
published: "2026-06-25T17:29:48.000Z"
score: 53
channel: "ai"
tags: ["Blackwell可逼近性","优化理论","在线学习","梯度均衡"]
---

# Blackwell可逼近性与梯度均衡的等价性研究

## 速览

本研究证明了梯度均衡（GEQ）与Blackwell可逼近性在算法意义上是等价的。GEQ是近年来提出的在线优化框架，可推广离线优化中的一阶平稳性，并抽象出在线共形预测等问题。通过构造双向归约，论文揭示了GEQ与遗憾最小化、校准等经典在线学习框架的深层联系，并提出了GEQ的充要条件，以及无约束与约束决策集下不同GEQ概念间的归约方法。该工作为在线学习理论的统一提供了关键环节。

## 关键点

- 梯度均衡（GEQ）与Blackwell可逼近性在算法意义上完全等价，可相互归约且无渐近损失。
- 通过已知的可逼近性与遗憾最小化、校准的等价性，GEQ也等价于这些框架。
- 提出了GEQ的充要条件，并建立了无约束与约束决策集下不同GEQ概念间的归约。
- 归约过程高效，可将遗憾最小化中的优化和强自适应性等精细保证转移到GEQ。

## 全文

Gradient equilibrium (GEQ) is a recently introduced online optimization framework that generalizes first-order stationarity from offline optimization and abstracts problems like online conformal prediction. While GEQ has curious similarities with known online learning frameworks, namely regret minimization, prior work has shown that GEQ error and regret are incomparable objectives, leaving open a precise understanding of how GEQ fits into the broader online learning landscape. In this work, we show that GEQ is equivalent to Blackwell approachability in the algorithmic sense. That is, a Blackwell approachability problem can always be solved using queries to a black-box GEQ oracle, with no asymptotic loss in the oracle's error rate, and vice versa. Taken together with known equivalences between approachability, regret minimization, and calibration, these results imply that GEQ is equivalent to these frameworks, as well. Our reductions are efficient and can be used to transfer refined guarantees, such as optimism and strong adaptivity, from regret minimization to GEQ. Along the way, we also identify necessary and sufficient conditions for GEQ, and establish reductions between different notions of GEQ with unconstrained and constrained decision sets.

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引用前请核对原文;本文由星际流动整理:https://feed.astralor.com/items/itm_a460bc7beaee6fd7867f6851