Optimal anytime regret with two experts
This solves a long-standing open problem in online learning for the two-expert case, providing the first minimax optimal algorithm in the anytime setting.
The paper tackles the problem of determining the minimax optimal regret for prediction with expert advice in the anytime setting with two experts, proving that the optimal regret is γ√t/2 at all time steps t, where γ is a known constant from Brownian motion.
We consider the classical problem of prediction with expert advice. In the fixed-time setting, where the time horizon is known in advance, algorithms that achieve the optimal regret are known when there are two, three, or four experts or when the number of experts is large. Much less is known about the problem in the anytime setting, where the time horizon is not known in advance. No minimax optimal algorithm was previously known in the anytime setting, regardless of the number of experts. Even for the case of two experts, Luo and Schapire have left open the problem of determining the optimal algorithm. We design the first minimax optimal algorithm for minimizing regret in the anytime setting. We consider the case of two experts, and prove that the optimal regret is $γ\sqrt{t} / 2$ at all time steps $t$, where $γ$ is a natural constant that arose 35 years ago in studying fundamental properties of Brownian motion. The algorithm is designed by considering a continuous analogue of the regret problem, which is solved using ideas from stochastic calculus.