Performance of Multiplicative-weights-updates

Let us remember the mechanics of Multiplicative-Weights-Updates: At every time t, the learner maintains a weight vector wt ≥ 0 over the experts. Given the weight vector, the probability distribution over the experts is computed as pt = wt wt·1 . The weights are initialized at w1 = 1 n · 1. (Multiplicative-weights-update step.) Given the loss vector at time t the weights are updated as follows wt+1(i) = wt(i) · uβ(lt(i)),∀i, where uβ : [0, 1]→ [0, 1] is an update function satisfying β ≤ uβ(x) ≤ 1− (1− β)x, ∀x ∈ [0, 1], for some β ∈ [0, 1]. The reader is free to chose whatever function uβ s/he wants. For example, one can use uβ(x) = β , for any β ∈ [0, 1]. In this case, wt+1(i) = wt(i) · βt = . . . = w1(i) · βt ≡ 1 nβ t.