Finite-Time 4-Expert Prediction Problem
This provides a theoretical solution for a specific case in online learning, but it is incremental as it extends prior conjectures to N=4.
The paper solves the finite-time expert prediction problem for N=4 experts by explicitly solving the nonlinear PDE from dynamic programming, proving that conjectured strategies form an asymptotic Nash equilibrium and validating the 'Finite vs Geometric regret' conjecture.
We explicitly solve the nonlinear PDE that is the continuous limit of dynamic programming of \emph{expert prediction problem} in finite horizon setting with $N=4$ experts. The \emph{expert prediction problem} is formulated as a zero sum game between a player and an adversary. By showing that the solution is $\mathcal{C}^2$, we are able to show that the strategies conjectured in arXiv:1409.3040G form an asymptotic Nash equilibrium. We also prove the "Finite vs Geometric regret" conjecture proposed in arXiv:1409.3040G for $N=4$, and and show that this conjecture in fact follows from the conjecture that the comb strategies are optimal.