Blackwell Approachability and Gradient Equilibrium are Equivalent

2026-06-25

AI Summary

This work clarifies the theoretical standing of Gradient Equilibration (GEQ) in online learning by establishing, for the first time, an algorithmic equivalence between GEQ and Blackwell approachability through bidirectional, computationally efficient reductions. Building on this equivalence, the paper further demonstrates that GEQ is fundamentally equivalent to other central frameworks such as regret minimization and calibration. The analysis integrates tools from Blackwell’s theory, online optimization, reduction techniques, and oracle-based query mechanisms, while carefully distinguishing between constrained and unconstrained decision sets. By solidifying GEQ’s foundational role, the study enables reductions among various GEQ variants and facilitates the transfer of refined properties—such as optimism and strong adaptivity—from regret minimization into the GEQ framework, thereby enriching its theoretical guarantees.

Abstract

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.

Problem

• Gradient Equilibrium
• Blackwell Approachability
• Online Learning
• Regret Minimization
• Calibration

Innovation

• Gradient Equilibrium
• Blackwell Approachability
• Regret Minimization
• Online Learning
• Algorithmic Equivalence