# A PDE Approach to the Prediction of a Binary Sequence with Advice from Two History-Dependent Experts

arXiv: https://arxiv.org/abs/2007.12732 · HTML: https://mlsanity.com/paper/2007.12732/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Jul 2020.

## Built on by (most cited first, 7 of 7)

- [PDE-Based Optimal Strategy for Unconstrained Online Learning](https://mlsanity.com/paper/2201.07877/) (2022)
- [A PDE-Based Analysis of the Symmetric Two-Armed Bernoulli Bandit](https://mlsanity.com/paper/2202.05767/) (2022)
- [Numerical solution of a PDE arising from prediction with expert advice](https://mlsanity.com/paper/2406.05754/) (2024)
- [An explicit solution of the five-expert prediction PDE and the exact optimality set of COMB](https://mlsanity.com/paper/2609.14892/) (2026)
- [A PDE approach for regret bounds under partial monitoring](https://mlsanity.com/paper/2209.01256/) (2022)
- [Reasoning without Regret](https://mlsanity.com/paper/2504.09777/) (2025)
- [On prediction from expert advice with more than five experts](https://mlsanity.com/paper/2610.05186/) (2026)

## Abstract (arXiv)

The prediction of a binary sequence is a classic example of online machine learning. We like to call it the 'stock prediction problem,' viewing the sequence as the price history of a stock that goes up or down one unit at each time step. In this problem, an investor has access to the predictions of two or more 'experts,' and strives to minimize her final-time regret with respect to the best-performing expert. Probability plays no role; rather, the market is assumed to be adversarial. We consider the case when there are two history-dependent experts, whose predictions are determined by the d most recent stock moves. Focusing on an appropriate continuum limit and using methods from optimal control, graph theory, and partial differential equations, we discuss strategies for the investor and the adversarial market, and we determine associated upper and lower bounds for the investor's final-time regret. When d is less than 4 our upper and lower bounds coalesce, so the proposed strategies are asymptotically optimal. Compared to other recent applications of partial differential equations to prediction, ours has a new element: there are two timescales, since the recent history changes at every step whereas regret accumulates more slowly.