A Conservation Law for Equilibrium Propagation and Coupled Learning

Physical learning rules such as equilibrium propagation (EP), coupled learning (CL), and adjoint coupled learning (AL) train resistive networks through local measurements. In the small-nudge limit EP and CL exactly conserve the conductance mass K = (1/2) sum_e kappa_e^2, a property that stabilizes training. We show that conservation also governs the inductive bias of these rules. For a single output we prove that EP and CL are trajectory equivalent, so single-output experiments cannot distinguish what the two rules learn. We prove that AL does not conserve the mass but dissipates it at exactly twice its own loss. In linear circuits we prove that the conserved mass has no functional consequence: all three vector fields are homogeneous in the conductances, so the selected solution is independent of the initialization scale. Fixed nonlinear elements break this protection. In diode circuits the learned input-output function depends on the initialization scale by up to about forty percent, an effect absent in linear controls, and the conservative rules retain this memory permanently while the dissipative rule partially erases it. At matched training loss the dissipative rule typically generalizes worse than the conservative rules, although it reaches low training loss faster; the penalty correlates with the mass dissipated en route and fades in larger circuits, where little mass is lost. The conservation structure of a local learning rule thus sets its initialization memory, its training speed, and, where dissipation is appreciable, its generalization; it should be treated as a design parameter of physical learning machines.

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.

Physical learning lets a trainable material or network use its own physical response to carry error signals, reducing the need for a separately programmed backward computation. We ask what determines whether such a system follows conventional gradient descent or evolves along a genuinely different learning trajectory. Our canonical model is a directed layered transport network in which every node redistributes a fixed amount of flow, so learning preserves positivity and total mass. In this model, conservation constrains only the allowable learning directions. Within the matched response class studied here, adjoint matching gives the physical output response a symmetric form. Non-negative mode-wise feedback then produces a reciprocal closed-loop response and a reweighted gradient flow. Adding an antisymmetric boundary component makes the closed-loop response rotational: the learning path can turn while the error driving that update still decreases at that moment. Turning is not automatically beneficial. Its finite-step effect is set by local curvature, and its accumulated effect also depends on step selection and on the new states visited along the path. Numerical consistency checks reproduce the exact response structure, predict the sign of the local effect across new network families, and show how trajectory drift can negate a local advantage. These results separate the roles of conservation, reciprocity, and nonreciprocity in physical learning.