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Equilibrium propagation (EP) is an alternative to backpropagation (BP) thatallows the training of deep neural networks with local learning rules. It thusprovides a compelling framework for training neuromorphic systems andunderstanding learning in neurobiology. However, EP requires infinitesimalteaching signals, thereby limiting its applicability in noisy physical systems.Moreover, the algorithm requires separate temporal phases and has not beenapplied to large-scale problems. Here we address these issues by extending EPto holomorphic networks. We show analytically that this extension naturallyleads to exact gradients even for finite-amplitude teaching signals.Importantly, the gradient can be computed as the first Fourier coefficient fromfinite neuronal activity oscillations in continuous time without requiringseparate phases. Further, we demonstrate in numerical simulations that ourapproach permits robust estimation of gradients in the presence of noise andthat deeper models benefit from the finite teaching signals. Finally, weestablish the first benchmark for EP on the ImageNet 32x32 dataset and showthat it matches the performance of an equivalent network trained with BP. Ourwork provides analytical insights that enable scaling EP to large-scaleproblems and establishes a formal framework for how oscillations could supportlearning in biological and neuromorphic systems.

Holomorphic Equilibrium Propagation Computes Exact Gradients Through Finite Size Oscillations