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Event-Driven Contrastive Divergence for Spiking Neuromorphic Systems
Emre Neftci, Srinjoy Das, Bruno Pedroni, Kenneth Kreutz-Delgado, Gert Cauwenberghs
TL;DR
The paper addresses the difficulty of mapping discrete, exact-arithmetic CD training onto continuous-time neuromorphic systems. It uses neural sampling with recurrent I&F dynamics and STDP-based event-driven updates, achieving MNIST recognition close to standard CD while supporting generative-model functionality.
Problem
RBM training relies on batch, discrete-time updates and exact arithmetic that do not directly map onto continuous-time neuromorphic neural substrates.
Method
The method uses I&F spiking neurons whose recurrent activity samples the RBM distribution and whose STDP synapses implement online, asynchronous CD updates.
Results
The spiking RBM learned a generative MNIST model with recognition accuracy up to 91.9%, versus 93.6% for standard CD and Gibbs sampling.
Takeaways & Limitations
Event-driven CD enables the same dynamical system to support training and testing while carrying out recognition, generation, and cue-integration tasks.
Takeaways & Limitations
Sampling performance degrades when hardware-realistic alpha-shaped PSPs replace rectangular PSPs, and generative performance requires several neurons per class label rather than one.
Abstract
from arXiv · showhide
Restricted Boltzmann Machines (RBMs) and Deep Belief Networks have been demonstrated to perform efficiently in a variety of applications, such as dimensionality reduction, feature learning, and classification. Their implementation on neuromorphic hardware platforms emulating large-scale networks of spiking neurons can have significant advantages from the perspectives of scalability, power dissipation and real-time interfacing with the environment. However the traditional RBM architecture and the commonly used training algorithm known as Contrastive Divergence (CD) are based on discrete updates and exact arithmetics which do not directly map onto a dynamical neural substrate. Here, we present an event-driven variation of CD to train a RBM constructed with Integrate & Fire (I&F) neurons, that is constrained by the limitations of existing and near future neuromorphic hardware platforms. Our strategy is based on neural sampling, which allows us to synthesize a spiking neural network that samples from a target Boltzmann distribution. The recurrent activity of the network replaces the discrete steps of the CD algorithm, while Spike Time Dependent Plasticity (STDP) carries out the weight updates in an online, asynchronous fashion. We demonstrate our approach by training an RBM composed of leaky I&F neurons with STDP synapses to learn a generative model of the MNIST hand-written digit dataset, and by testing it in recognition, generation and cue integration tasks. Our results contribute to a machine learning-driven approach for synthesizing networks of spiking neurons capable of carrying out practical, high-level functionality.
1 Introduction
RBMs are attractive targets for neuromorphic hardware because their distributed structure may support scalable, low-power, real-time computation. The paper addresses the mismatch between digitally defined CD training and continuous-time spiking dynamics with an online, event-driven approach.
- Motivation: RBM and deep-network training often requires substantial computational resources on clusters or GPUs, motivating dedicated low-power, scalable hardware.Neuromorphic implementations are presented as appealing alternatives for these resource-intensive models.
- Neuromorphic platform: Neuromorphic VLSI emulates densely packed biological neurons in real time while dissipating very low power.Its asynchronous address-event communication can also interface with event-based sensors and support scalable implementations.
- Problem: Digital RBMs and CD use batch, discrete-time iterative updates and exact arithmetic, whereas neuromorphic systems compute through continuous-time I&F dynamics.This mismatch makes transferring conventional training algorithms to neuromorphic substrates non-trivial.
- Approach: The paper constructs RBMs with I&F neurons and trains them using an online, event-driven adaptation of CD.The approach draws on neural sampling to represent the RBM’s probability distribution with spiking activity.
- Approach: Recurrent activity replaces CD’s discrete construction and reconstruction steps, while STDP updates weights online and asynchronously from individual spikes.Over longer periods, these microscopic updates reproduce macroscopic CD updates without separate training-time connection reprogramming.
- Demonstration: 91.9% recognition accuracy was achieved on MNIST, compared with 93.6% for standard CD and Gibbs sampling.The network used 824 visible and 500 hidden neurons and learned a generative model.
2 Materials and Methods
The paper constructs neural samplers from stochastic leaky I&F neurons whose recurrent activity represents RBM sampling, while alpha-shaped PSPs and parameter calibration adapt the model to hardware. The resulting sampler approximates Boltzmann distributions, with accuracy limited by the I&F and synaptic dynamics.
- Neural sampling conditions: Neural sampling requires stochastic firing rates and membrane potentials equal to linear sums of synaptic inputs, yielding a network that samples a Boltzmann distribution.The abstract neuron model uses spike-triggered binary states lasting for the refractory period.
- Hardware-compatible synapses: Alpha-shaped PSPs replace rectangular, non-additive PSPs because linear synaptic filters are more practical for hardware implementation.This substitution is identified as the major source of degradation in sampling performance.
- Noisy I&F realization: Leaky I&F neurons with noisy currents realize the abstract sampling model approximately through their input-output transfer curve.The transfer curve is analyzed from the Langevin dynamics using the Fokker–Planck equation.
- Noisy I&F realization: Reducing the ratio (θ−urst)/σV improves the fit between the noisy I&F transfer curve and a sigmoid function for a fixed refractory period.The fit improves as the threshold-to-reset separation becomes smaller relative to voltage noise.
- Calibration and stability: Background noise must dominate fluctuations from network inputs so the fitted parameters β and γ remain approximately constant during sampling.Otherwise, changes in input-current variance effectively scale synaptic weights and biases.
3 Results
The paper implements RBM training on spiking neuromorphic systems by replacing discrete Contrastive Divergence updates with online, asynchronous STDP driven by neural sampling. On MNIST, the resulting I&F RBM supports recognition while retaining robustness to parameter discretization and shorter training.
- An RBM contains visible and hidden layers with symmetric interlayer connections but no within-layer connections.
- Event-Driven Contrastive Divergence: Event-driven CD uses neural sampling and spike-triggered, asynchronous synaptic updates to implement CD online in neuromorphic hardware.The rule uses a modulated, pair-based STDP mechanism whose average behavior approximates the macroscopic CD update.
- Event-Driven Contrastive Divergence: A zero-mean modulatory signal switches synapses between LTP and LTD, separating data and reconstruction phases while recurrent activity replaces discrete CD steps.The signal changes slowly relative to network dynamics, and a refractory-period delay suppresses transient effects after phase transitions.
- Learning a generative model of hand-written digits: 91.9% recognition accuracy was achieved by an I&F RBM trained with event-driven CD, compared with 93.6% for standard CD and Gibbs sampling.The event-driven model also reached 90.8% when evaluated by free-energy minimization, indicating consistency with the RBM description.
- Learning a generative model of hand-written digits: 20,000 data presentations, corresponding to 2,000 s of simulated time, were sufficient for event-driven CD training versus 250,000 presentations for offline training.The authors state that these training durations are achievable for real-time neuromorphic systems.
- Learning a generative model of hand-written digits: Recognition degraded gracefully under parameter discretization, reaching 91.6% with 8-bit parameters and 89.4% with 5-bit parameters.Both discretized models retained discriminative power, although the 5-bit reduction caused a larger decline.
4 Discussion
The paper frames event-driven CD as a way to train spiking RBMs online while preserving the same connectivity during training and testing. It demonstrates neural-sampling extensions, hardware robustness considerations, practical MNIST functionality, and remaining architectural and learning-rule limitations.
- Event-driven learning: Event-driven CD uses recurrent activity and STDP dynamics to replace discrete CD updates, keeping network connectivity unchanged during training and testing.This supports online learning while retaining recognition, data generation, and cue integration functionality.
- Neural sampling: The neural-sampling framework interprets spikes as samples and extends prior work to I&F sampling conditions, finite-precision robustness, and hidden-unit learning with STDP synapses.These extensions provide the sampling basis for the spiking RBM training method.
- Hardware limitations: Large-scale deployment remains constrained by unsatisfied RBM symmetry, roughly 2 million recurrent synapses, and the need to test sparse interlayer connectivity.RRAM crossbars are discussed as a possible solution for synapse implementation, while sparse connectivity remains untested.
- Learning-rule limitations: The method is functional but not known to be optimal because CDk is approximate, I&F sampling is not exactly Boltzmann, and event-driven updates differ from standard CD updates.The paper identifies custom plasticity rules minimizing functionally relevant objectives as future research.