# A continuous-time approach to online optimization

arXiv: https://arxiv.org/abs/1401.6956 · HTML: https://mlsanity.com/paper/1401.6956/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Jan 2014.

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

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- [Online Optimization in Games via Control Theory: Connecting Regret, Passivity and Poincar'e Recurrence](https://mlsanity.com/paper/2106.04748/) (2021)
- [Evolutionary Dynamics and Phi-Regret Minimization in Games](https://mlsanity.com/paper/2106.14668/) (2021)
- [The impact of uncertainty on regularized learning in games](https://mlsanity.com/paper/2506.13286/) (2025)
- [Constant regret in general games via higher-order optimism](https://mlsanity.com/paper/2609.04113/) (2026)
- [Optimal Anytime Algorithms for Online Convex Optimization with Adversarial Constraints](https://mlsanity.com/paper/2510.22579/) (2025)
- [Smooth Quadratic Prediction Markets](https://mlsanity.com/paper/2505.02959/) (2025)
- [No Coin Left Behind: Maximizing Strategic Surplus Against No-Regret Dynamics](https://mlsanity.com/paper/2604.05129/) (2026)
- [Regret, equilibrium, and learning in games: A guided tour](https://mlsanity.com/paper/2608.09389/) (2026)

## Abstract (arXiv)

We consider a family of learning strategies for online optimization problems that evolve in continuous time and we show that they lead to no regret. From a more traditional, discrete-time viewpoint, this continuous-time approach allows us to derive the no-regret properties of a large class of discrete-time algorithms including as special cases the exponential weight algorithm, online mirror descent, smooth fictitious play and vanishingly smooth fictitious play. In so doing, we obtain a unified view of many classical regret bounds, and we show that they can be decomposed into a term stemming from continuous-time considerations and a term which measures the disparity between discrete and continuous time. As a result, we obtain a general class of infinite horizon learning strategies that guarantee an O(n-1/2) regret bound without having to resort to a doubling trick.