分析物理学习在线性电路中的局部收敛，揭示收敛关键条件。
Coercivity and Local Convergence of Physical Learning in Linear Circuits
- 基于网络关联结构构造矩阵，提出可强制性条件判定收敛性。
- 小扰动下，损失指数下降，参数收敛至解流形，但对称电路可能失效。
- 适用于研究物理启发学习算法的理论性质，如神经形态计算开发者。
物理学习方法通过仅使用局部更新规则，利用系统物理特性实现全局信息传递，训练物理网络完成计算任务。本文首次对三种方法——平衡传播（EP）、耦合学习（CL）和新提出的伴随耦合学习（AL）——在线性电路中进行了局部收敛分析，涵盖离散与连续时间情形，且在小扰动极限下成立。EP与AL在自然损失函数上执行梯度下降，而CL遵循带三次修正的修正动力学。假设解存在，我们提出一个由网络关联结构构建的矩阵秩条件作为强制性条件，该条件下训练损失指数衰减，参数收敛至解流形。通过构造风筝电路示例，表明对称性可能导致解流形上强制性常数退化；但利用萨尔德定理证明，此类退化非典型：对几乎所有期望输出，强制性在解流形上处处成立。
原文摘要 · Abstract (English)
Physical learning methods train physical networks to perform computational tasks using only local update rules, exploiting the physics of the system to handle the global transfer of information. We provide the first local convergence analysis of three such methods -- Equilibrium Propagation (EP), Coupled Learning (CL), and a new method we call Adjoint Coupled Learning (AL) -- for linear circuits, in the limit of small-nudging for both discrete and continuous time. EP and AL perform gradient descent on a natural loss function, while CL follows modified dynamics with an additional cubic correction. Assuming the existence of a solution, we identify a coercivity condition, expressed as a rank condition on a matrix built from the network's incidence structure, under which the training loss decays exponentially and the parameters converge to the solution manifold. We show that coercivity can fail by exhibiting a kite circuit in which a symmetry causes the coercivity constant to degenerate on the solution manifold, but prove using Sard's theorem that such degeneracies are non-generic: coercivity holds at every point of the solution manifold for almost every choice of desired output.
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