Lagrangian-based Equilibrium Propagation: generalisation to arbitrary boundary conditions & equivalence with Hamiltonian Echo Learning (2506.06248v1)

Abstract: Equilibrium Propagation (EP) is a learning algorithm for training Energy-based Models (EBMs) on static inputs which leverages the variational description of their fixed points. Extending EP to time-varying inputs is a challenging problem, as the variational description must apply to the entire system trajectory rather than just fixed points, and careful consideration of boundary conditions becomes essential. In this work, we present Generalized Lagrangian Equilibrium Propagation (GLEP), which extends the variational formulation of EP to time-varying inputs. We demonstrate that GLEP yields different learning algorithms depending on the boundary conditions of the system, many of which are impractical for implementation. We then show that Hamiltonian Echo Learning (HEL) -- which includes the recently proposed Recurrent HEL (RHEL) and the earlier known Hamiltonian Echo Backpropagation (HEB) algorithms -- can be derived as a special case of GLEP. Notably, HEL is the only instance of GLEP we found that inherits the properties that make EP a desirable alternative to backpropagation for hardware implementations: it operates in a "forward-only" manner (i.e. using the same system for both inference and learning), it scales efficiently (requiring only two or more passes through the system regardless of model size), and enables local learning.

• The paper introduces GLEP as a novel extension to Equilibrium Propagation that accommodates time-varying inputs using action principles.
• It formulates various boundary conditions (CIVP, CBVP, PFVP) to enhance computational efficiency in neuromorphic and analog systems.
• The analysis establishes equivalence with Hamiltonian Echo Learning via Legendre transformation, paving the way for real-time adaptive learning.

Analyzing the Extensions of Equilibrium Propagation using Generalized Lagrangian Formulations

The paper "Lagrangian-based Equilibrium Propagation: generalisation to arbitrary boundary conditions content equivalence with Hamiltonian Echo Learning" explores significant advancements in Equilibrium Propagation (EP) for learning algorithms in the context of ai and neural networks. It introduces Generalized Lagrangian Equilibrium Propagation (GLEP) as an extension of EP to address the challenges associated with time-varying inputs, which is a crucial necessity for systems that traditionally rely on static input-output mappings.

Overview and Key Contributions

The central idea of the paper revolves around broadening the application of EP, a framework promising local learning rules and unbiased gradient estimates without requiring explicit backpropagation. EP is grounded in energy-based models where learning occurs via perturbations to natural system dynamics. However, EP traditionally presupposes static environments, limiting its scope for dynamically evolving systems—a constraint this paper aims to overcome.

The paper methodically presents GLEP, a theoretical framework leveraging action principles instead of the prevalent energy minimization techniques. This shift enables the formulation of learning dynamics across temporal domains by redefining learning algorithms through augmented action functionals rather than scalar energy functions. Notably, GLEP is complemented by the introduction of various boundary conditions affecting the viability and feasibility of the resulting learning algorithms.

The authors discuss several boundary conditions—Constant Initial Value Problem (CIVP), Constant Boundary Value Problem (CBVP), and Parametric Final Value Problem (PFVP). Each formulation distinctly influences computational efficiency and practical implementation, serving as pivotal elements in translating theory into accessible technology for neuromorphic and analog computing architectures.

The robustness of GLEP is consolidated through theoretical analysis revealing how Hamiltonian Echo Learning (HEL), including its variant Recurrent HEL (RHEL), can be derived from specific implementations of GLEP. This connection is demonstrated using the Legendre transformation, establishing equivalence between Lagrangian-based and Hamiltonian-based approaches to computation and learning. The findings underscore the potential of GLEP not just as an abstract concept but as a cornerstone for practical learning systems extending beyond traditional digital computers.

Implications and Future Directions

The paper posits that the successful integration of GLEP into hardware implementations could be transformative for fields reliant on reinforcement learning in dynamic environments, such as robotics and autonomous systems. Importantly, the work hints at developing learning paradigms that align more closely with biological neural processing, advancing beyond the limitations posed by backpropagation.

For future research, the authors suggest focusing on:

• Developing online variants of RHEL to eliminate dependency on retrospective echo phases.
• Extending GLEP frameworks for broader application beyond strictly reversible systems, enhancing flexibility for general-purpose neural circuits.

In essence, the presented research offers a promising theoretical advancement with potential practical ramifications in designing energy-efficient, real-time adaptive systems capable of handling time-varying inputs in both artificial intelligence and computational neuroscience. The unified theory of learning algorithms blending action-based and Hamiltonian dynamics signifies a pivotal shift, potentially heralding new research directions and methodologies in complex system training and deployment.