The rate regions of multi-terminal Gaussian channels (multiple-access, broadcast, interference, relay) are delimited by conditional mutual informations $I(V_A;V_B\,|\,V_C)$ among groups of input and output nodes; bringing such channels under differentiable physical-layer design therefore hinges on evaluating any such conditional MI, and its gradient, on a unified computation graph. Modeling the network as a linear Gaussian directed acyclic graph (Gaussian-DAG), we obtain $I(V_A;V_B\,|\,V_C)$ in closed form: from the node-pair covariances produced by one K-recursion forward pass, it is a log-determinant difference of two sub-block Schur complements of the support covariance. The construction is built entirely from automatic-differentiation (AD) primitives, so any differentiable function of finitely many conditional MIs is end-to-end differentiable in the design parameters; this broad class includes linear objectives (weighted sum-rate, secrecy), the rate functions of standard multi-terminal rate regions, and non-linear composites of these. A single reverse-mode AD sweep yields the Wirtinger gradient with respect to all controllable factors at once, so any such objective can be handled by projected gradient iterations without problem-specific gradient derivation. We demonstrate the framework on three experiments: rate-region maximization for a two-user MIMO multiple-access channel, secure precoding on a MIMO wiretap channel, and the same rate-region objective applied to a larger multi-hop multiple-access network.

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