这篇论文把梯度均衡、Blackwell可逼近性和遗憾最小化这几个在线学习的关键框架串起来了，证明它们本质等价，对理解在线优化底层逻辑很有帮助。
该论文证明了梯度均衡（GEQ）与Blackwell可逼近性在算法上是等价的：任何Blackwell可逼近问题可通过调用GEQ黑盒求解，且误差率无渐进损失；反之亦然。结合已知的Blackwell可逼近性与遗憾最小化、校准之间的等价关系，GEQ也被证明与这些框架等价。论文还给出了高效归约，可将乐观性和强自适应性等精细保证从遗憾最小化迁移到GEQ。此外，作者识别出GEQ的充要条件，并建立了无约束与有约束决策集下不同GEQ概念之间的归约。
Blackwell Approachability and Gradient Equilibrium are Equivalent
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.