Online Linear Optimization with Sparsity

Now, let us consider the case ofK = Kb with b ∈ (1,∞). For v ∈ R andQ ⊆ [d], let vQ denote the 6 projection of v to those dimensions inQ. Then for any v ∈ R, and any w ∈ Kb withQ = {i : wi 6= 7 0}, we know by Hölder’s inequality that 〈w,v〉 = 〈wQ,vQ〉 ≥ −‖w‖b · ‖vQ‖a , for a = b/(b− 1). 8 Moreover, one can have 〈wQ,vQ〉 = −‖w‖b · ‖vQ‖a , when |wi| /‖w‖b = |vi|/‖v‖a and 9 wivi ≤ 0 for every i ∈ Q. Thus, to find w ∈ Kb which minimizes 〈w,v〉, we first letQ = [k] so that 10 ‖vQ‖a ≥ ‖vQ′‖a for any Q ′ ⊆ [d] with |Q′| ≤ k. Then we let ŵi = −sign(vi)|vi|1i∈Q, and 11 choose wi = ŵi/‖ŵ‖b. Clearly, we have w ∈ K and 〈w,v〉 = −‖vQ‖a ≤ −‖vQ′‖a ≤ 〈w ′,v〉 12 for any w′ ∈ K with Q′ = {i : w′ i 6= 0} as |Q′| ≤ k. 13