# A PDE approach for regret bounds under partial monitoring

arXiv: https://arxiv.org/abs/2209.01256 · HTML: https://mlsanity.com/paper/2209.01256/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 29 references that are in the map.
- First posted on arXiv 2 Sep 2022.

## Builds on (most cited first, 16 of 16)

- [PDE-Based Optimal Strategy for Unconstrained Online Learning](https://mlsanity.com/paper/2201.07877/) (2022)
- [Prediction with Expert Advice: a PDE Perspective](https://mlsanity.com/paper/1904.11401/) (2019)
- [New Potential-Based Bounds for Prediction with Expert Advice](https://mlsanity.com/paper/1911.01641/) (2019)
- [Optimal anytime regret with two experts](https://mlsanity.com/paper/2002.08994/) (2020)
- [Finite-Time 4-Expert Prediction Problem](https://mlsanity.com/paper/1911.10936/) (2019)
- [Towards Optimal Algorithms for Prediction with Expert Advice](https://mlsanity.com/paper/1409.3040/) (2014)
- [Optimal Non-Asymptotic Lower Bound on the Minimax Regret of Learning with Expert Advice](https://mlsanity.com/paper/1511.02176/) (2015)
- [On the asymptotic optimality of the comb strategy for prediction with expert advice](https://mlsanity.com/paper/1902.02368/) (2019)
- [Tight Lower Bounds for Multiplicative Weights Algorithmic Families](https://mlsanity.com/paper/1607.02834/) (2016)
- [A PDE Approach to the Prediction of a Binary Sequence with Advice from Two History-Dependent Experts](https://mlsanity.com/paper/2007.12732/) (2020)
- [Malicious Experts versus the multiplicative weights algorithm in online prediction](https://mlsanity.com/paper/2003.08457/) (2020)
- [Efficient and Optimal Fixed-Time Regret with Two Experts](https://mlsanity.com/paper/2203.07577/) (2022)
- [A PDE-Based Analysis of the Symmetric Two-Armed Bernoulli Bandit](https://mlsanity.com/paper/2202.05767/) (2022)
- [Viscosity solutions for controlled McKean--Vlasov jump-diffusions](https://mlsanity.com/paper/1909.12337/) (2019)
- [Master Bellman equation in the Wasserstein space: Uniqueness of viscosity solutions](https://mlsanity.com/paper/2107.10535/) (2021)
- [Randomized filtering and Bellman equation in Wasserstein space for partial observation control problem](https://mlsanity.com/paper/1609.02697/) (2016)

## Abstract (arXiv)

In this paper, we study a learning problem in which a forecaster only observes partial information. By properly rescaling the problem, we heuristically derive a limiting PDE on Wasserstein space which characterizes the asymptotic behavior of the regret of the forecaster. Using a verification type argument, we show that the problem of obtaining regret bounds and efficient algorithms can be tackled by finding appropriate smooth sub/supersolutions of this parabolic PDE.