We establish sharp asymptotically optimal strategies for the problem of online prediction with history dependent experts. The prediction problem is played (in part) over a discrete graph called the d dimensional de Bruijn graph, where d is the number of days of history used by the experts. Previous work Drenska and Kohn (arXiv:2007.12732, 2020) established O(ε) optimal strategies for n= 2 experts and d≤ 4 days of history, while Drenska and Kohn (J Nonlinear Sci 30. 30(1), 137–173, 2020) established O(ε1 / 3) optimal strategies for all n≥ 2 and all d≥ 1 , where the game is played for N steps and ε= N- 1 / 2. In this paper, we show that the optimality conditions over the de Bruijn graph correspond to a graph Poisson equation, and we establish O(ε) optimal strategies for all values of n and d.
Bibliographical noteFunding Information:
Funding Jeff Calder was supported by NSF-DMS Grant 1944925 and the Alfred P. Sloan foundation.
© 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC part of Springer Nature.
- De Bruijn graph
- Graph Laplacian
- Partial differential equations
- Poisson equation
- Prediction with expert advice
- Viscosity solutions