a compact set X ⊆ RD . - We treat f as a blackbox which has no closed-form, no derivatives, non-convex etc. - That only allow us to query its function value at ∀x ∈ X. - We address a global optimization problem : x* = argmax x∈X f(x) . - To ensure that a global optimum is found, we require good coverage of X. - As the dimensionality increases, the number of evaluations needed to cover X increases exponentially. 3
R - d e < D : eﬀective dimensionality - T ⊂ RD : a linear subspace, dim(T) = d e - T⊥ ⊂ RD : orthogonal complement of T - If we have f(x) = f(xT + x⊥) = f(xT), xT ∈ T , x⊥ ∈ T⊥ then we call T the eﬀective subspace and T⊥ constant space. 6
|| 2 ≤ (√d e / ε) || x T * || 2 with probability at least 1 - ε REMBO algorithm 8 The box constraints : X = [-1, 1]D , always available through rescaling. In all experiments in this paper, y ∈ Y = [-√d, √d]d . Is this part time consuming ?