Generalizing equilibrium propagation to Lagrangian systems with arbitrary boundary conditions & equivalence with Hamiltonian echo learning

Abstract

Equilibrium Propagation (EP) is a learning algorithm that applies to Energy-Based Models (EBMs) on static inputs. It estimates loss gradients by contrasting two steady states of the same EBM, rather than resorting to explicit adjoint dynamics. EP originally appealed as a plausible learning theory for biological substrates and has more recently attracted interest for its amenability to analog hardware. Extending EP to time-varying inputs and outputs is a challenging problem, as the variational description must apply to the entire system trajectory rather than just its steady state. While the use of the action of a Lagrangian system as an energy function appears as a natural choice—which we herein refer to as Lagrangian Equilibrium Propagation (LEP)—careful consideration of boundary conditions was largely overlooked in prior studies, although it becomes essential. It is also unclear how applying LEP to Lagrangian systems theoretically relates to applying Hamiltonian Echo Learning (HEL) algorithms -i.e. Hamiltonian Echo Backpropagation (HEB) and Recurrent Hamiltonian Echo Learning (RHEL)—to Hamiltonian systems. In this work, we thoroughly revisit LEP and demonstrate that different learning algorithms can be obtained depending on the boundary conditions of the system, many of which are impractical to simulate—e.g., with a prohibitive memory or computational cost or requiring explicit Jacobian computation. We also show that HEL algorithms, which are much easier to simulate, can be explicitly cast as a special case of LEP where the initial conditions can be picked arbitrarily. Building upon this connection enables the extension of LEP to a broader class of systems with dissipation terms. By filtering out intractable instantiations of LEP and building an explicit mapping between HEL and LEP algorithms, this work facilitates the simulation of self-learning Lagrangian systems as well as extensions of LEP to broader classes of physical systems.