Gradient-free training of recurrent neural networks (2410.23467v1)

Abstract: Recurrent neural networks are a successful neural architecture for many time-dependent problems, including time series analysis, forecasting, and modeling of dynamical systems. Training such networks with backpropagation through time is a notoriously difficult problem because their loss gradients tend to explode or vanish. In this contribution, we introduce a computational approach to construct all weights and biases of a recurrent neural network without using gradient-based methods. The approach is based on a combination of random feature networks and Koopman operator theory for dynamical systems. The hidden parameters of a single recurrent block are sampled at random, while the outer weights are constructed using extended dynamic mode decomposition. This approach alleviates all problems with backpropagation commonly related to recurrent networks. The connection to Koopman operator theory also allows us to start using results in this area to analyze recurrent neural networks. In computational experiments on time series, forecasting for chaotic dynamical systems, and control problems, as well as on weather data, we observe that the training time and forecasting accuracy of the recurrent neural networks we construct are improved when compared to commonly used gradient-based methods.

• The paper introduces a novel gradient-free training approach that bypasses backpropagation using random parameter sampling and EDMD.
• It leverages Koopman operator theory to linearize nonlinear dynamics, mitigating the exploding and vanishing gradient issues.
• Empirical results show reduced model fitting times with competitive accuracy on chaotic systems and real-world weather data.

Gradient-Free Training of Recurrent Neural Networks

The paper "Gradient-Free Training of Recurrent Neural Networks" by Erik Lien Bolager et al. presents a novel methodology for constructing and training Recurrent Neural Networks (RNNs) without relying on gradient-based methods. This approach is primarily motivated by the challenges posed by the Exploding and Vanishing Gradient Problem (EVGP) inherent in traditional gradient-based training methods for RNNs, which can lead to instability and inefficiencies in training, particularly for time-dependent problems such as time series analysis and forecasting.

The core of the proposed technique involves a non-traditional approach to setting network weights and biases. Instead of adjusting these parameters through backpropagation, the method employs random sampling for the hidden layer parameters and Extended Dynamic Mode Decomposition (EDMD) to compute the outer weights. This is aligned with the Koopman operator theory, which approximates a linear mapping of nonlinear dynamical systems. This connection potentially melds the robustness of linear systems theory with the flexibility of RNNs, promising stable and efficient training outcomes.

The paper thoroughly grounds this methodology in mathematical foundations, defining a framework based on sampling and Koopman operator theory. By circumventing the direct computation of gradients, the proposed method sidesteps many traditional RNN training issues, including those exacerbated by chaotic dynamics and bifurcations that are not easily addressed even with advanced RNN architectures such as LSTMs and GRUs.

In empirical evaluations, the authors test this method against traditional gradient-based approaches, such as the shallow PLRNN (shPLRNN), and reservoir computing models like Echo State Networks (ESNs), across several datasets, including synthetic data from canonical chaotic systems and real-world weather data. Key findings from these experiments reveal that the gradient-free approach significantly reduces model fitting time while providing comparable or better performance in terms of accuracy across various evaluation metrics like mean squared error (MSE) and empirical Kullback-Leibler divergence (EKL).

One of the notable advantages of this method is its potential for faster model development cycles due to reduced computational requirements in hyperparameter tuning, driven by fewer parameters needing adjustment compared with traditional RNNs. Additionally, the convergence proofs provided for the gradient-free training approach in infinite network width contexts are well-founded, establishing a rigorous basis for the practical applicability of this technique.

Implications of this research stretch into theoretical and practical realms: theoretically, it opens up new avenues to explore the synthesis of theoretical systemic frameworks like the Koopman operator within the neural network discipline; practically, the significantly enhanced training efficiency positions this methodology as a potent tool for real-time systems and applications needing low-latency predictions.

Future work could refine and extend this gradient-free method to controlled dynamical systems, where direct applications to further complex system modeling and diverse domains could be explored. Additionally, bridging these concepts to continuously running systems such as NeuralODEs offers an intriguing pathway for future research, broadening the adaptable applications of these methodologies.

In conclusion, the paper meticulously presents a novel framework for training RNNs, coupling computational efficiency with sophisticated mathematical structures, making this approach both practical for applications demanding fast solutions and robust enough to handle complex, unpredictable dynamics.