aiqbench analysis
This research investigates how physical constraints like conservation laws and reciprocity alter learning trajectories in trainable networks, distinguishing them from standard gradient descent. The study reveals that while rotational components in the response can locally reduce error, their cumulative effect depends heavily on step selection and trajectory drift, challenging the assumption that such dynamics are inherently beneficial. These findings advance the theoretical understanding of adaptive physical systems where hardware properties directly encode learning signals without separate backward computation.
Reciprocity Separates Gradient Flow from Rotation in Conservative Physical Learning
arXiv:2608.30778v1 Announce Type: new Abstract: 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.
Abstract: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.
Machine Learning (cs.LG); Adaptation and Self-Organizing Systems (nlin.AO)
arXiv:2608.30778 [cs.LG]
(or arXiv:2608.30778v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2608.30778 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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