Mutual Information Constrained Chernoff Bottleneck

The classical information bottleneck (IB) measures the relevance of a representation $U$ of $X$ to a target $Y$ by $I(U;Y)$, which does not directly characterize the error of downstream decisions. For a binary hypothesis $Y$ inferred from many separately encoded observations, the optimal error exponent is the Chernoff information between the two conditional distributions of $U$ given $Y$. We study the mutual information constrained Chernoff bottleneck, which seeks an encoder that maximizes this Chernoff information subject to a rate constraint $I(U;X) \leq R$. We show that its optimal value $C(R)$ increases strictly up to $R = H(V)$, where $V$ merges the symbols of $X$ with equal likelihood ratio, remains at the uncompressed exponent beyond, and, unlike the IB curve, need not be concave. We further show that $k+1$ outputs suffice to attain $C(R)$, where $k$ is the cardinality of $V$. We propose an alternating algorithm that updates the encoder via a generalized Blahut--Arimoto algorithm and the Chernoff parameter $s$ via a nonlinear equation, and prove that its iterates remain feasible, with nondecreasing and convergent Chernoff information. Comparing with the classical IB for $n$ samples, we show that, whenever the two objectives disagree, the IB encoder has the smaller equivocation $H(Y|U^{n})$ at $n = 1$, whereas the Chernoff bottleneck has the smaller error probability and equivocation for all large $n$. Numerical experiments confirm the theory, and on real topic-detection data from the 20 Newsgroups corpus, compressing each word to only $17\%$ of its entropy retains $90\%$ of the error exponent and nearly the accuracy of the uncompressed classifier.