Mutual Information Constrained Chernoff Bottleneck

Abstract: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. 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.

31 pages, 3 figures, 2 tables. Feedback and comments are welcome

Information Theory (cs.IT); Machine Learning (cs.LG); Optimization and Control (math.OC); Statistics Theory (math.ST); Machine Learning (stat.ML)

E.4; H.1.1; G.3; I.2.6

arXiv:2609.37994 [cs.IT]

(or arXiv:2609.37994v1 [cs.IT] for this version)

https://doi.org/10.48550/arXiv.2609.37994 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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