Equilibrium propagation (EP) is a compelling alternative to the backpropagation 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 estimate unbiased gradients efficiently. Both requirements are challenging to implement in physical systems. Yet, whether and how weight asymmetry affects its applicability is unknown because, in practice, it may be masked by biases introduced through the 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 the finite nudge does not pose a problem, as exact derivatives can still be estimated via a Cauchy integral. In contrast, weight asymmetry introduces bias resulting in low task performance due to poor alignment of EP's neuronal error vectors compared to BP. To mitigate this 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 32x32. 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.

翻译：平衡传播（EP）是误差反向传播算法（BP）的一种有力替代方案，适用于生物或类神经形态底物上神经网络的梯度计算。然而，该算法需要权重对称性以及无限小的平衡摄动（即微扰）来高效估计无偏梯度。这两个要求在物理系统中难以实现。由于实际中有限微扰引入的偏差可能掩盖权重不对称性的影响，目前尚不清楚权重不对称性是否及如何影响其适用性。为解答这一问题，我们研究可无需权重对称性表述的广义平衡传播，并通过分析将两种偏差源分离。对于复可微的非对称网络，我们证明有限微扰不会造成问题，因为可通过柯西积分估计精确导数。相反，权重不对称性会引入偏差，导致EP的神经元误差向量与BP的对齐性差，从而降低任务性能。为缓解此问题，我们提出一种新的稳态目标函数，直接惩罚网络固定点处雅可比矩阵的功能不对称性。该稳态目标函数显著提升了网络解决复杂任务（如ImageNet 32x32）的能力。我们的结果奠定了理论研究基础，有助于理解和缓解物理网络缺陷对依赖底物松弛动力学的学习算法带来的不利影响。