Benjamin Scellier

Siddhartha Mishra

Yann Ollivier

2022-05-30

ArXiv (preprint)

Scaling Equilibrium Propagation to Deep ConvNets by Drastically Reducing Its Gradient Estimator Bias

Axel Laborieux

Maxence Ernoult

Julie Grollier

Damien Querlioz

2021-02-18

Frontiers in Neuroscience (published)

Training End-to-End Analog Neural Networks with Equilibrium Propagation

Jack D. Kendall

Ross D. Pantone

Kalpana Manickavasagam

We introduce a principled method to train end-to-end analog neural networks by stochastic gradient descent. In these analog neural networks,… (see more) the weights to be adjusted are implemented by the conductances of programmable resistive devices such as memristors [Chua, 1971], and the nonlinear transfer functions (or `activation functions') are implemented by nonlinear components such as diodes. We show mathematically that a class of analog neural networks (called nonlinear resistive networks) are energy-based models: they possess an energy function as a consequence of Kirchhoff's laws governing electrical circuits. This property enables us to train them using the Equilibrium Propagation framework [Scellier and Bengio, 2017]. Our update rule for each conductance, which is local and relies solely on the voltage drop across the corresponding resistor, is shown to compute the gradient of the loss function. Our numerical simulations, which use the SPICE-based Spectre simulation framework to simulate the dynamics of electrical circuits, demonstrate training on the MNIST classification task, performing comparably or better than equivalent-size software-based neural networks. Our work can guide the development of a new generation of ultra-fast, compact and low-power neural networks supporting on-chip learning.

2020-06-02

ArXiv (preprint)

Continual Weight Updates and Convolutional Architectures for Equilibrium Propagation

Maxence Ernoult

Julie Grollier

Damien Querlioz

Equilibrium Propagation (EP) is a biologically inspired alternative algorithm to backpropagation (BP) for training neural networks. It appli… (see more)es to RNNs fed by a static input x that settle to a steady state, such as Hopfield networks. EP is similar to BP in that in the second phase of training, an error signal propagates backwards in the layers of the network, but contrary to BP, the learning rule of EP is spatially local. Nonetheless, EP suffers from two major limitations. On the one hand, due to its formulation in terms of real-time dynamics, EP entails long simulation times, which limits its applicability to practical tasks. On the other hand, the biological plausibility of EP is limited by the fact that its learning rule is not local in time: the synapse update is performed after the dynamics of the second phase have converged and requires information of the first phase that is no longer available physically. Our work addresses these two issues and aims at widening the spectrum of EP from standard machine learning models to more bio-realistic neural networks. First, we propose a discrete-time formulation of EP which enables to simplify equations, speed up training and extend EP to CNNs. Our CNN model achieves the best performance ever reported on MNIST with EP. Using the same discrete-time formulation, we introduce Continual Equilibrium Propagation (C-EP): the weights of the network are adjusted continually in the second phase of training using local information in space and time. We show that in the limit of slow changes of synaptic strengths and small nudging, C-EP is equivalent to BPTT (Theorem 1). We numerically demonstrate Theorem 1 and C-EP training on MNIST and generalize it to the bio-realistic situation of a neural network with asymmetric connections between neurons.

2020-04-29

ArXiv (preprint)

A deep learning framework for neuroscience

Blake Richards

Timothy P. Lillicrap

Philippe Beaudoin

Rafal Bogacz

Amelia Christensen

Claudia Clopath

Rui Ponte Costa

Archy de Berker

Surya Ganguli

Colleen J Gillon

Danijar Hafner

Adam Kepecs

Nikolaus Kriegeskorte

Peter Latham

Grace W. Lindsay

Kenneth D. Miller

Richard Naud

Christopher C. Pack

Panayiota Poirazi … (see 12 more)

Pieter Roelfsema

João Sacramento

Andrew Saxe

Anna C. Schapiro

Walter Senn

Greg Wayne

Daniel Yamins

Friedemann Zenke

Joel Zylberberg

Denis Therien

Konrad Paul Kording

2019-10-28

Nature Neuroscience (published)

Equivalence of Equilibrium Propagation and Recurrent Backpropagation

Recurrent backpropagation and equilibrium propagation are supervised learning algorithms for fixed-point recurrent neural networks, which di… (see more)ffer in their second phase. In the first phase, both algorithms converge to a fixed point that corresponds to the configuration where the prediction is made. In the second phase, equilibrium propagation relaxes to another nearby fixed point corresponding to smaller prediction error, whereas recurrent backpropagation uses a side network to compute error derivatives iteratively. In this work, we establish a close connection between these two algorithms. We show that at every moment in the second phase, the temporal derivatives of the neural activities in equilibrium propagation are equal to the error derivatives computed iteratively by recurrent backpropagation in the side network. This work shows that it is not required to have a side network for the computation of error derivatives and supports the hypothesis that in biological neural networks, temporal derivatives of neural activities may code for error signals.

2019-02-01

Neural Computation (published)

openreview.net

Updates of Equilibrium Prop Match Gradients of Backprop Through Time in an RNN with Static Input

Maxence Ernoult

Julie Grollier

Damien Querlioz

Equilibrium Propagation (EP) is a biologically inspired learning algorithm for convergent recurrent neural networks, i.e. RNNs that are fed … (see more)by a static input x and settle to a steady state. Training convergent RNNs consists in adjusting the weights until the steady state of output neurons coincides with a target y. Convergent RNNs can also be trained with the more conventional Backpropagation Through Time (BPTT) algorithm. In its original formulation EP was described in the case of real-time neuronal dynamics, which is computationally costly. In this work, we introduce a discrete-time version of EP with simplified equations and with reduced simulation time, bringing EP closer to practical machine learning tasks. We first prove theoretically, as well as numerically that the neural and weight updates of EP, computed by forward-time dynamics, are step-by-step equal to the ones obtained by BPTT, with gradients computed backward in time. The equality is strict when the transition function of the dynamics derives from a primitive function and the steady state is maintained long enough. We then show for more standard discrete-time neural network dynamics that the same property is approximately respected and we subsequently demonstrate training with EP with equivalent performance to BPTT. In particular, we define the first convolutional architecture trained with EP achieving ~ 1% test error on MNIST, which is the lowest error reported with EP. These results can guide the development of deep neural networks trained with EP.

Generalization of Equilibrium Propagation to Vector Field Dynamics

The biological plausibility of the backpropagation algorithm has long been doubted by neuroscientists. Two major reasons are that neurons wo… (see more)uld need to send two different types of signal in the forward and backward phases, and that pairs of neurons would need to communicate through symmetric bidirectional connections. We present a simple two-phase learning procedure for fixed point recurrent networks that addresses both these issues. In our model, neurons perform leaky integration and synaptic weights are updated through a local mechanism. Our learning method generalizes Equilibrium Propagation to vector field dynamics, relaxing the requirement of an energy function. As a consequence of this generalization, the algorithm does not compute the true gradient of the objective function, but rather approximates it at a precision which is proven to be directly related to the degree of symmetry of the feedforward and feedback weights. We show experimentally that our algorithm optimizes the objective function.

2018-08-14

ArXiv (preprint)