# Blackwell Approachability and Gradient Equilibrium are Equivalent

arXiv: https://arxiv.org/abs/2606.27315 · HTML: https://mlsanity.com/paper/2606.27315/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 26 references that are in the map.
- First posted on arXiv 25 Jun 2026.

## Builds on (most cited first, 14 of 14)

- [Online Learning: A Modern Introduction Using Convex Optimization](https://mlsanity.com/paper/1912.13213/) (2019)
- [Optimization, Learning, and Games with Predictable Sequences](https://mlsanity.com/paper/1311.1869/) (2013)
- [Strongly Adaptive Online Learning](https://mlsanity.com/paper/1502.07073/) (2015)
- [Improved Strongly Adaptive Online Learning using Coin Betting](https://mlsanity.com/paper/1610.04578/) (2016)
- [Near-Optimal Algorithms for Omniprediction](https://mlsanity.com/paper/2501.17205/) (2025)
- [A Unifying Perspective on Multi-Calibration: Game Dynamics for Multi-Objective Learning](https://mlsanity.com/paper/2302.10863/) (2023)
- [Online Minimax Multiobjective Optimization: Multicalibeating and Other Applications](https://mlsanity.com/paper/2108.03837/) (2021)
- [Forecast Hedging and Calibration](https://mlsanity.com/paper/2210.07169/) (2022)
- [Approachability, Regret and Calibration; implications and equivalences](https://mlsanity.com/paper/1301.2663/) (2013)
- [Gradient Equilibrium in Online Learning: Theory and Applications](https://mlsanity.com/paper/2501.08330/) (2025)
- [Faster Recalibration of an Online Predictor via Approachability](https://mlsanity.com/paper/2310.17002/) (2023)
- [An Online Convex Optimization Approach to Blackwell's Approachability](https://mlsanity.com/paper/1503.00255/) (2015)
- [Panprediction: Optimal Predictions for Any Downstream Task and Loss](https://mlsanity.com/paper/2510.27638/) (2025)
- [Calibrated Multi-Level Quantile Forecasting](https://mlsanity.com/paper/2512.23671/) (2025)

## Abstract (arXiv)

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.