Forward Direct Feedback Alignment for Online Gradient Estimates of Spiking Neural Networks

TL;DR

The paper tackles energy-efficient training of Spiking Neural Networks on neuromorphic hardware by proposing Spiking Forward Direct Feedback Alignment (SFDFA), an online adaptation of Forward Direct Feedback Alignment that uses fixed feedback connections to estimate output–hidden weights. It derives exact local gradients for spike timings, accounts for intra-neuron dependencies, and introduces a hardware-friendly eligibility-trace formulation that avoids backpropagation through time. Empirically, SFDFA improves convergence and gradient/weight alignment relative to DFA and closely matches BP on static datasets, though a gap remains on highly temporal data like SHD. The work demonstrates a viable path toward neuromorphic gradient descent with direct feedback and identifies temporal-data challenges and future directions for further closing the gap with BP.

Abstract

There is an interest in finding energy efficient alternatives to current state of the art neural network training algorithms. Spiking neural network are a promising approach, because they can be simulated energy efficiently on neuromorphic hardware platforms. However, these platforms come with limitations on the design of the training algorithm. Most importantly, backpropagation cannot be implemented on those. We propose a novel neuromorphic algorithm, the \textit{Spiking Forward Direct Feedback Alignment} (SFDFA) algorithm, an adaption of \textit{Forward Direct Feedback Alignment} to train SNNs. SFDFA estimates the weights between output and hidden neurons as feedback connections. The main contribution of this paper is to describe how exact local gradients of spikes can be computed in an online manner while taking into account the intra-neuron dependencies between post-synaptic spikes and derive a dynamical system for neuromorphic hardware compatibility. We compare the SFDFA algorithm with a number of competitor algorithms and show that the proposed algorithm achieves higher performance and convergence rates.

Figures (10)

• Figure 1: Original factor $\frac{\tau}{x - \vartheta}$ (red line) and the modified factor $\frac{\tau}{x}$ as a function of $x=a_i^k \exp\left(\frac{-t_i^k}{\tau_s}\right)$. As $a_i^k \exp\left(\frac{-t_i^k}{\tau_s}\right)$ approaches $\vartheta$, the original factor diverges towards infinity while the modified factor is bounded.
• Figure 2: Gradient fields of a single neuron with two inputs computed using the exact local gradients \ref{['fig:grad_field_original']} and the modified local gradients \ref{['fig:grad_field_modified']}. We can observe that the exact local gradient contains critical points where the norm is abnormally large compared to neighboring regions. However, with the modified local gradients, the norm of the gradient is consistent throughout the weight space, mitigating the gradient explosion caused by the critical points. The red crosses at correspond to the critical point visualized in Figure \ref{['fig:instability']}. See main text for details on how this figure was generated. For these simulations, we used $\tau_s=0.01$, $\theta=0.01$ and the time window was of length $0.01$ seconds.
• Figure 3: Evolution over time of the membrane potential \ref{['fig:instability_mp']}, input current \ref{['fig:instability_current']}, gradient factors \ref{['fig:instability_factors']} and local gradients \ref{['fig:instability_loc_gradient']} at an instability point ($w_{i,1}=2.5$ and $w_{i,2}=2.0$ in Figure \ref{['fig:grad_fields']}) in a minimal example of a single spiking neuron with two inputs. Here all vertical dotted lines correspond to post-synaptic spike times. The red lines in both \ref{['fig:instability_factors']} and \ref{['fig:instability_loc_gradient']} represent the original factor and local gradient, while the blue lines represent the modified factor and local gradient. These figures show that the last post-synaptic spike fired by the neuron occurs when the membrane potential narrowly reaches the threshold. This narrow threshold crossing is attributed to the low input current $I_i(t)$ which causes large factors $\frac{\tau}{I_i(t) - \vartheta}$ and consequently large local gradients. In contrast, our modified factor $\frac{\tau}{I_i(t)}$ restricts the amplitude of the factor, thereby moderating the scale of the gradient. For these simulations, we used $\tau_s=0.01$, $\theta=0.01$ and the time window was of length $0.01$ seconds.
• Figure 4: Evolution of the training accuracy (Figure \ref{['fig:factor_mnist_train_acc']}), test accuracy (Figure \ref{['fig:factor_mnist_test_acc']}), train loss (Figure \ref{['fig:factor_mnist_train_loss']}) and test loss (Figure \ref{['fig:factor_mnist_test_loss']}) during the training of a two-layers SNN on the MNIST dataset. Red lines correspond to the metrics of the SNN updated using the exact local gradient while blue lines correspond to the metrics of the SNN updated using the modified local gradient defined in Equation \ref{['eq:loc_grad_fixed']}. These figures show that the modified gradient converges slightly faster than the exact local gradient.
• Figure 5: Evolution of the averaged train accuracy (Figure \ref{['fig:sfdfa_mnist_train_acc']}), test accuracy (Figure \ref{['fig:sfdfa_mnist_test_acc']}), train loss (Figure \ref{['fig:sfdfa_mnist_train_loss']}) and test loss (Figure \ref{['fig:sfdfa_mnist_test_loss']}) during the training of two-layers SNNs on the MNIST dataset. Black dashed lines correspond to BP. Blue and red solid lines correspond to the SFDFA and DFA algorithms respectively.