Does regret change when the loss function is dependent on the previous predictions?


The loss function of each expert in the expert advice problem(or any online learning problem) depends on the time($ t$ ) and expert advice at that time($ f_{t}(i)$ ). suppose in this problem, loss function depends on the previous prediction of the algorithm. $ $ l _{t} (i) = p_{1} p_{2} \cdots p_{t-1}f_{t}(i)$ $ such that $ p$ show prediction of algorithm.

Does the upper bound of regret change?