Learning long range dependencies through time reversal symmetry breaking

Abstract: Deep State Space Models (SSMs) reignite physics-grounded compute paradigms, as RNNs could natively be embodied into dynamical systems. This calls for dedicated learning algorithms obeying to core physical principles, with efficient techniques to simulate these systems and guide their design. We propose Recurrent Hamiltonian Echo Learning (RHEL), an algorithm which provably computes loss gradients as finite differences of physical trajectories of non-dissipative, Hamiltonian systems. In ML terms, RHEL only requires three "forward passes" irrespective of model size, without explicit Jacobian computation, nor incurring any variance in the gradient estimation. Motivated by the physical realization of our algorithm, we first introduce RHEL in continuous time and demonstrate its formal equivalence with the continuous adjoint state method. To facilitate the simulation of Hamiltonian systems trained by RHEL, we propose a discrete-time version of RHEL which is equivalent to Backpropagation Through Time (BPTT) when applied to a class of recurrent modules which we call Hamiltonian Recurrent Units (HRUs). This setting allows us to demonstrate the scalability of RHEL by generalizing these results to hierarchies of HRUs, which we call Hamiltonian SSMs (HSSMs). We apply RHEL to train HSSMs with linear and nonlinear dynamics on a variety of time-series tasks ranging from mid-range to long-range classification and regression with sequence length reaching $\sim 50k$. We show that RHEL consistently matches the performance of BPTT across all models and tasks. This work opens new doors for the design of scalable, energy-efficient physical systems endowed with self-learning capabilities for sequence modelling.

• The paper presents RHEL, an algorithm that leverages time reversal symmetry breaking to compute gradients efficiently in Hamiltonian neural systems.
• It details continuous and discrete implementations, demonstrating equivalence with adjoint state methods and competitive performance to BPTT.
• RHEL scales to hierarchical state space models, effectively handling extensive sequences of up to 50,000 time steps in classification and regression tasks.

Learning Long-Range Dependencies through Time Reversal Symmetry Breaking

The paper "Learning Long Range Dependencies through Time Reversal Symmetry Breaking" by Guillaume Pourcel and Maxence Ernoult addresses the challenge of effectively training recurrent neural networks (RNNs) when embodied as dynamical physical systems. To tackle this problem, the authors propose Recurrent Hamiltonian Echo Learning (RHEL), an algorithm inspired by Hamiltonian Echo Backpropagation (HEB) that allows for efficient gradient computation in non-dissipative Hamiltonian systems without requiring explicit state-Jacobian computations.

Overview

RHEL is introduced as a novel approach that computes gradients using trajectories from Hamiltonian systems, leveraging physical principles such as time-reversal symmetry. The algorithm requires only three forward passes to estimate loss gradients across RNNs, avoiding direct backward computation and thereby reducing computational complexity. This approach is particularly beneficial for handling long-range dependencies within sequence modeling tasks, where data sequence lengths can reach as high as 50,000 time steps.

Core Contributions

• Continuous and Discrete Implementations: The paper describes RHEL in both continuous and discrete time frameworks. In continuous time, it is proven to be equivalent to the continuous adjoint state method in the limit of small trajectory perturbations. The discrete-time implementation shows that RHEL can achieve the effectiveness of Backpropagation Through Time (BPTT) in a class termed Hamiltonian Recurrent Units (HRUs).
• Hierarchical Learning: RHEL is scalable across hierarchical state space models, termed Hamiltonian State Space Models (HSSMs), that stack recurrent units. This hierarchical arrangement enables RHEL to handle complex sequential data modeling tasks with its learning dynamics.
• Application to Real-World Tasks: The paper demonstrates the efficacy of RHEL by applying it to train HSSMs on several time-series tasks, including classification and regression. The algorithm consistently performs on par with BPTT across various tests, asserting its competitiveness in a practical context.

Numerical Insights

The numerical evaluations highlight the capability of RHEL to closely align with BPTT in gradient estimation accuracy, as showcased through tasks involving diverse datasets with varying sequence lengths. The linear and nonlinear dynamics trained using RHEL form robust models capable of performing accurate classification and regression over extensive time-series datasets.

Implications and Future Work

RHEL presents promising implications for the development of scalable, energy-efficient physical systems in sequence modeling. The algorithm connects machine learning with novel compute paradigms, suggesting practical applications in domains reliant on long-range sequence dependencies like speech processing and complex predictive modeling.

Future developments might involve extending the RHEL framework to include truly dissipative systems, enhancing its flexibility and potential for broader model expressivity. Additionally, simplifying its integration with analog physical systems like photonic and spintronic circuits could leverage its underlying principles for drastic improvements in computational efficiency and scalability.

In summary, this research provides an innovative approach to recurrent model training through an intersection of machine learning and physical computation, inviting continued exploration and integration of this algorithm into practical applications and future artificial intelligence system designs.