# Gradient Equilibrium in Online Learning: Theory and Applications

arXiv: https://arxiv.org/abs/2501.08330 · HTML: https://mlsanity.com/paper/2501.08330/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Jan 2025.

## Built on by (most cited first, 13 of 13)

- [The Relationship between No-Regret Learning and Online Conformal Prediction](https://mlsanity.com/paper/2502.10947/) (2025)
- [Conformal Risk Control for Non-Monotonic Losses](https://mlsanity.com/paper/2602.20151/) (2026)
- [Mirror Online Conformal Prediction with Intermittent Feedback](https://mlsanity.com/paper/2503.10345/) (2025)
- [Prediction-Powered Communication with Distortion Guarantees](https://mlsanity.com/paper/2509.24373/) (2025)
- [Proximal Regret and Proximal Correlated Equilibria: A New Tractable Solution Concept for Online Learning and Games](https://mlsanity.com/paper/2511.01852/) (2025)
- [Distribution-informed Online Conformal Prediction](https://mlsanity.com/paper/2512.07770/) (2025)
- [Distribution free M-estimation](https://mlsanity.com/paper/2505.22807/) (2025)
- [Is Online Linear Optimization Sufficient for Strategic Robustness?](https://mlsanity.com/paper/2602.12253/) (2026)
- [Efficient Online Conformal Selection with Limited Feedback](https://mlsanity.com/paper/2605.14953/) (2026)
- [Online Conformal Prediction with Corrupted Feedback](https://mlsanity.com/paper/2605.20515/) (2026)
- [Parameter-Free and Group Conditional Online Conformal Prediction](https://mlsanity.com/paper/2606.00419/) (2026)
- [Blackwell Approachability and Gradient Equilibrium are Equivalent](https://mlsanity.com/paper/2606.27315/) (2026)
- [Paradoxes of Game Theoretic Equilibria and Price of Anarchy](https://mlsanity.com/paper/2607.11752/) (2026)

## Abstract (arXiv)

We present a new perspective on online learning that we refer to as gradient equilibrium: a sequence of iterates achieves gradient equilibrium if the average of gradients of losses along the sequence converges to zero. In general, this condition is not implied by, nor implies, sublinear regret. It turns out that gradient equilibrium is achievable by standard online learning methods such as gradient descent and mirror descent with constant step sizes (rather than decaying step sizes, as is usually required for no regret). Further, as we show through examples, gradient equilibrium translates into an interpretable and meaningful property in online prediction problems spanning regression, classification, quantile estimation, and others. Notably, we show that the gradient equilibrium framework can be used to develop a debiasing scheme for black-box predictions under arbitrary distribution shift, based on simple post hoc online descent updates. We also show that post hoc gradient updates can be used to calibrate predicted quantiles under distribution shift, and that the framework leads to unbiased Elo scores for pairwise preference prediction.