Generalizing Bottleneck Problems

Given a pair of random variables (X, Y) ~ P_XY and two convex functions f₁ and f₂, we introduce two bottleneck functionals as the lower and upper boundaries of the two-dimensional convex set that consists of the pairs (I_f1(W;X), I_f2(W;Y)), where If denotes f-information and W varies over the set of all discrete random variables satisfying the Markov condition W → X → Y. Applying Witsenhausen and Wyner's approach, we provide an algorithm for computing boundaries of this set for f₁, f₂, and discrete P_XY. In the binary symmetric case, we fully characterize the set when (i) f₁(t)=f₂(t)=tlogt, (ii) f₁(t)=-f₂(t)=t² - 1, and (iii) f₁ and f₂ are both l^β norm function for β ≥ 2. We then argue that upper and lower boundaries in (i) correspond to Mrs. Gerber's Lemma and its inverse (which we call Mr. Gerber's Lemma), in (ii) correspond to estimation-theoretic variants of Information Bottleneck and Privacy Funnel, and in (iii) correspond to Arimoto Information Bottleneck and Privacy Funnel.