Online algorithm for aggregating experts’ predictions with unbounded quadratic loss

We consider the problem of online aggregation of experts’ predictions with a quadratic loss function. At the beginning of each round t = 1, 2, . . . , T , experts n = 1, . . . , N provide predictions γ t , . . . , γ t ∈ H (where H is a Hilbert space). The player aggregates the predictions to a single prediction γt ∈ H. Then nature provides the true outcome ω ∈ H. The player and the experts n = 1, . . . , N suffer the losses ht = ∥ω−γt∥ and l t = ∥ω−γ t ∥, respectively, and the next round t + 1 begins. The goal of the player is to minimize the regret, that is, the difference between the total loss of the player and the loss of the best expert: RT = ∑T t=1 ht −minn=1,...,N ∑T t=1 l n t .