Theoretical Foundations of Surrogate Gradient Learning in Spiking Neural Networks

Core Concepts

Surrogate gradients provide a theoretically well-founded solution for end-to-end training of stochastic spiking neural networks.

Abstract

The content provides a detailed analysis of the theoretical foundations of surrogate gradient (SG) learning in spiking neural networks (SNNs). It examines the relationship between SGs and two other theoretically well-grounded approaches: smoothed probabilistic models (SPMs) and stochastic automatic differentiation (stochAD).
The key insights are:

For single binary Perceptrons, SGs are equivalent to the gradients of the expected output in SPMs and the smoothed stochastic derivatives in the stochAD framework.

In multi-layer Perceptrons (MLPs), SPMs lack support for efficient gradient computation via automatic differentiation, whereas stochAD provides the missing theoretical basis for SGs in stochastic SNNs.

SGs introduce bias in deterministic MLPs and are not gradients of a surrogate loss function.

The analysis extends to spiking leaky integrate-and-fire (LIF) neurons, confirming that SGs correspond to smoothed stochastic derivatives in stochastic SNNs.

Empirical simulations demonstrate that SG-descent can successfully train stochastic SNNs, preserving trial-to-trial variability, and outperforming deterministic SNNs in some cases.

In summary, the content establishes the theoretical foundation for the widely used SG learning method in the context of
stochastic SNNs, providing a rigorous justification for its empirical success.

Stats

"Training spiking neural networks to approximate complex functions is essential for studying information processing in the brain and neuromorphic computing."
"Surrogate gradients have proven empirically successful, but their theoretical foundation remains elusive."
"We find that the latter [stochAD] provides the missing theoretical basis for surrogate gradients in stochastic spiking neural networks."
"Surrogate gradients are not conservative fields and, thus, not gradients of a surrogate loss."

Quotes

"Surrogate gradients have proven empirically successful, but their theoretical foundation remains elusive."
"We find that the latter [stochAD] provides the missing theoretical basis for surrogate gradients in stochastic spiking neural networks."
"Surrogate gradients are not conservative fields and, thus, not gradients of a surrogate loss."

Key Insights Distilled From

Elucidating the theoretical underpinnings of surrogate gradient learning in spiking neural networks

by Julia Gygax,... at arxiv.org 04-24-2024

https://arxiv.org/pdf/2404.14964.pdf

Deeper Inquiries

How can the theoretical insights from this work be extended to other types of discrete neural network models beyond spiking neural networks

The theoretical insights from this work can be extended to other types of discrete neural network models beyond spiking neural networks by considering the fundamental principles of surrogate gradient learning. The concept of surrogate gradients, which provide a continuous relaxation of non-differentiable activation functions, can be applied to various types of discrete neural networks. For example, in binary neural networks or networks with discrete random variables, where traditional backpropagation is not directly applicable, surrogate gradients can offer a practical solution for training. By understanding the relationship between surrogate gradients and smoothed stochastic derivatives, as demonstrated in this work, researchers can apply similar principles to train and optimize different types of discrete neural network models. This extension can help address the challenges of non-differentiability in various neural network architectures and improve training efficiency and effectiveness.

What are the potential implications of the finding that surrogate gradients are not gradients of a surrogate loss function

The finding that surrogate gradients are not gradients of a surrogate loss function has significant implications for the training and optimization of neural networks. One implication is that the use of surrogate gradients introduces bias in the gradient approximation, which can impact the convergence and performance of the training process. This bias can lead to suboptimal solutions and affect the overall training dynamics of the network. Additionally, the fact that surrogate gradients do not correspond to gradients of a surrogate loss function means that the optimization process may not be following a well-defined objective function. This lack of alignment between the surrogate gradients and a surrogate loss function raises questions about the interpretability and reliability of the training process. Researchers and practitioners need to be aware of this limitation when using surrogate gradients in neural network training and consider potential adjustments or alternative approaches to mitigate the bias and ensure effective optimization.

How can the stochastic nature of spiking neural networks be further leveraged to improve their performance and biological plausibility compared to deterministic models

The stochastic nature of spiking neural networks can be leveraged to improve their performance and biological plausibility compared to deterministic models in several ways. Firstly, the inherent stochasticity in spiking neural networks can introduce variability in network dynamics, which can enhance robustness and adaptability to different stimuli and environments. By incorporating randomness in the spiking behavior of neurons, stochastic spiking neural networks can better capture the probabilistic nature of neural activity observed in biological systems. This can lead to more realistic modeling of neural processes and behaviors.
Furthermore, the stochastic nature of spiking neural networks can enable exploration of novel learning algorithms and optimization techniques that leverage randomness for improved performance. For example, techniques like stochastic gradient descent or reinforcement learning with exploration can be applied to train stochastic spiking neural networks more effectively. By introducing controlled randomness in the learning process, stochastic spiking neural networks can potentially achieve better generalization, faster convergence, and enhanced adaptability to changing conditions.
Overall, leveraging the stochastic nature of spiking neural networks can lead to advancements in both the computational efficiency and biological realism of neural network models, paving the way for more sophisticated and biologically plausible artificial intelligence systems.