Equilibrium propagation (EP) is a compelling alternative to the back propagation of error algorithm (BP) for computing gradients of neural networks on biological or analog neuromorphic substrates. Still, the algorithm requires weight symmetry and infinitesimal equilibrium perturbations, i.e., nudges, to yield unbiased gradient estimates.Both requirements are challenging to implement in physical systems.Yet, whether and how weight asymmetry contributes to bias is unknown because, in practice, its contribution may be masked by a finite nudge. To address this question, we study generalized EP, which can be formulated without weight symmetry, and analytically isolate the two sources of bias.For complex-differentiable non-symmetric networks, we show that bias due to finite nudge can be avoided by estimating exact derivatives via a Cauchy integral.In contrast, weight asymmetry induces residual bias  through poor alignment of EP's neuronal error vectors compared to BP resulting in low task performance.To mitigate the latter issue, we present a new homeostatic objective that directly penalizes functional asymmetries of the Jacobian at the network's fixed point. This homeostatic objective dramatically improves the network's ability to solve complex tasks such as ImageNet 32$\times$32. Our results lay the theoretical groundwork for studying and mitigating the adverse effects of imperfections of physical networks on learning algorithms that rely on the substrate's relaxation dynamics.
Improving equilibrium propagation without weight symmetry through Jacobian homeostasis
Axel Laborieux Friedemann Zenke