Coercivity and Local Convergence of Physical Learning in Linear Circuits

2026-06-13 · Source: Machine Learning · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Robotics & Autonomous Systems · Depth: Expert, quick

Summary

Coercivity and Local Convergence of Physical Learning in Linear Circuits" presents the first local convergence analysis for three physical learning methods: Equilibrium Propagation (EP), Coupled Learning (CL), and a new method called Adjoint Coupled Learning (AL). This analysis focuses on 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 uses modified dynamics with a cubic correction. The authors identify a coercivity condition, defined as a rank condition on a matrix derived from the network's incidence structure. Under this condition, the training loss decays exponentially, and parameters converge to the solution manifold. While a kite circuit demonstrates coercivity failure due to symmetry, Sard's theorem proves such degeneracies are non-generic for almost every desired output.

Key takeaway

For Research Scientists designing or implementing physical learning systems, understanding the coercivity condition is crucial for ensuring training stability and convergence. Your circuit's incidence structure directly impacts whether training loss will decay exponentially and parameters will converge. Prioritize network designs that satisfy this rank condition, and be aware that while specific symmetries can cause non-convergence, these are generally non-generic for desired outputs.

Key insights

This work provides the first local convergence analysis for physical learning methods in linear circuits under a coercivity condition.

Principles

• Physical learning leverages local rules for global information transfer.
• A coercivity condition ensures exponential loss decay and parameter convergence.
• Coercivity degeneracies are non-generic for most desired outputs.

Method

The analysis identifies a coercivity condition, expressed as a rank condition on a matrix from the network's incidence structure, to prove convergence for EP, CL, and AL in linear circuits.

In practice

• Design physical learning circuits to meet the identified coercivity condition.
• Be mindful of specific circuit symmetries that could cause coercivity failure.

Topics

• Physical Learning
• Linear Circuits
• Equilibrium Propagation
• Coupled Learning
• Adjoint Coupled Learning
• Convergence Analysis
• Optimization and Control

Best for: AI Scientist, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by Machine Learning.