Numerical solution of a PDE arising from prediction with expert advice
<|reference_start|>Numerical solution of a PDE arising from prediction with expert advice: This work investigates the online machine learning problem of prediction with expert advice in an adversarial setting through numerical analysis of, and experiments with, a related partial differential equation. The problem is a repeated two-person game involving decision-making at each step informed by $n$ experts in an adversarial environment. The continuum limit of this game over a large number of steps is a degenerate elliptic equation whose solution encodes the optimal strategies for both players. We develop numerical methods for approximating the solution of this equation in relatively high dimensions ($n\leq 10$) by exploiting symmetries in the equation and the solution to drastically reduce the size of the computational domain. Based on our numerical results we make a number of conjectures about the optimality of various adversarial strategies, in particular about the non-optimality of the COMB strategy.<|reference_end|>
@article{calder2024numerical,
    title={Numerical solution of a PDE arising from prediction with expert advice},
    author={Jeff Calder, Nadejda Drenska, Drisana Mosaphir},
    journal={arXiv preprint arXiv:2406.05754},
    year={2024},
    archivePrefix={arXiv},
    eprint={2406.05754},
    primaryClass={math.NA cs.AI cs.LG cs.NA math.AP}
}