Towards training digitally-tied analog blocks via hybrid gradient computation

Power efficiency is plateauing in the standard digital electronics realm such that novel hardware, models, and algorithms are needed to reduce the costs of AI training. The combination of energy-based analog circuits and the Equilibrium Propagation (EP) algorithm constitutes one compelling alternative compute paradigm for gradient-based optimization of neural nets. Existing analog hardware accelerators, however, typically incorporate digital circuitry to sustain auxiliary non-weight-stationary operations, mitigate analog device imperfections, and leverage existing digital accelerators.This heterogeneous hardware approach calls for a new theoretical model building block. In this work, we introduce Feedforward-tied Energy-based Models (ff-EBMs), a hybrid model comprising feedforward and energy-based blocks accounting for digital and analog circuits. We derive a novel algorithm to compute gradients end-to-end in ff-EBMs by backpropagating and "eq-propagating" through feedforward and energy-based parts respectively, enabling EP to be applied to much more flexible and realistic architectures. We experimentally demonstrate the effectiveness of the proposed approach on ff-EBMs where Deep Hopfield Networks (DHNs) are used as energy-based blocks. We first show that a standard DHN can be arbitrarily split into any uniform size while maintaining performance. We then train ff-EBMs on ImageNet32 where we establish new SOTA performance in the EP literature (46 top-1 %). Our approach offers a principled, scalable, and incremental roadmap to gradually integrate self-trainable analog computational primitives into existing digital accelerators.

Energy-based learning algorithms for analog computing: a comparative study

Energy-based learning algorithms have recently gained a surge of interest due to their compatibility with analog (post-digital) hardware. Existing algorithms include contrastive learning (CL), equilibrium propagation (EP) and coupled learning (CpL), all consisting in contrasting two states, and differing in the type of perturbation used to obtain the second state from the first one. However, these algorithms have never been explicitly compared on equal footing with same models and datasets, making it difficult to assess their scalability and decide which one to select in practice. In this work, we carry out a comparison of seven learning algorithms, namely CL and different variants of EP and CpL depending on the signs of the perturbations. Specifically, using these learning algorithms, we train deep convolutional Hopfield networks (DCHNs) on five vision tasks (MNIST, F-MNIST, SVHN, CIFAR-10 and CIFAR-100). We find that, while all algorithms yield comparable performance on MNIST, important differences in performance arise as the difficulty of the task increases. Our key findings reveal that negative perturbations are better than positive ones, and highlight the centered variant of EP (which uses two perturbations of opposite sign) as the best-performing algorithm. We also endorse these findings with theoretical arguments. Additionally, we establish new SOTA results with DCHNs on all five datasets, both in performance and speed. In particular, our DCHN simulations are 13.5 times faster with respect to Laborieux et al. (2021), which we achieve thanks to the use of a novel energy minimisation algorithm based on asynchronous updates, combined with reduced precision (16 bits).

Towards Scaling Difference Target Propagation by Learning Backprop Targets

The development of biologically-plausible learning algorithms is important for understanding learning in the brain, but most of them fail to scale-up to real-world tasks, limiting their potential as explanations for learning by real brains. As such, it is important to explore learning algorithms that come with strong theoretical guarantees and can match the performance of backpropagation (BP) on complex tasks. One such algorithm is Difference Target Propagation (DTP), a biologically-plausible learning algorithm whose close relation with Gauss-Newton (GN) optimization has been recently established. However, the conditions under which this connection rigorously holds preclude layer-wise training of the feedback pathway synaptic weights (which is more biologically plausible). Moreover, good alignment between DTP weight updates and loss gradients is only loosely guaranteed and under very specific conditions for the architecture being trained. In this paper, we propose a novel feedback weight training scheme that ensures both that DTP approximates BP and that layer-wise feedback weight training can be restored without sacrificing any theoretical guarantees. Our theory is corroborated by experimental results and we report the best performance ever achieved by DTP on CIFAR-10 and ImageNet 32$\times$32

Training Dynamical Binary Neural Networks with Equilibrium Propagation

Equilibrium Propagation (EP) is an algorithm intrinsically adapted to the training of physical networks, in particular thanks to the local updates of weights given by the internal dynamics of the system. However, the construction of such a hardware requires to make the algorithm compatible with the existing neuromorphic CMOS technology, which generally exploits digital communication between neurons and offers a limited amount of local memory. In this work, we demonstrate that EP can train dynamical networks with binary activations and weights. We first train systems with binary weights and full-precision activations, achieving an accuracy equivalent to that of full-precision models trained by standard EP on MNIST, and losing only 1.9% accuracy on CIFAR-10 with equal architecture. We then extend our method to the training of models with binary activations and weights on MNIST, achieving an accuracy within 1% of the full-precision reference for fully connected architectures and reaching the full-precision reference accuracy for the convolutional architecture. Our extension of EP training to binary networks is consistent with the requirements of today's dynamic, brain-inspired hardware platforms and paves the way for very low-power end-to-end learning.

Synaptic metaplasticity in binarized neural networks

Unlike the brain, artificial neural networks, including state-of-the-art deep neural networks for computer vision, are subject to "catastrophic forgetting": they rapidly forget the previous task when trained on a new one. Neuroscience suggests that biological synapses avoid this issue through the process of synaptic consolidation and metaplasticity: the plasticity itself changes upon repeated synaptic events. In this work, we show that this concept of metaplasticity can be transferred to a particular type of deep neural networks, binarized neural networks, to reduce catastrophic forgetting.

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

Equilibrium Propagation (EP) is a biologically-inspired counterpart of Backpropagation Through Time (BPTT) which, owing to its strong theoretical guarantees and the locality in space of its learning rule, fosters the design of energy-efficient hardware dedicated to learning. In practice, however, EP does not scale to visual tasks harder than MNIST. In this work, we show that a bias in the gradient estimate of EP, inherent in the use of finite nudging, is responsible for this phenomenon and that cancelling it allows training deep ConvNets by EP, including architectures with distinct forward and backward connections. These results highlight EP as a scalable approach to compute error gradients in deep neural networks, thereby motivating its hardware implementation.

EqSpike: Spike-driven Equilibrium Propagation for Neuromorphic Implementations

Neuromorphic systems achieve high energy efficiency by computing with spikes, in a brain-inspired way. However, finding spike-based learning algorithms that can be implemented within the local constraints of neuromorphic systems, while achieving high accuracy, remains a formidable challenge. Equilibrium Propagation is a hardware-friendly counterpart of backpropagation which only involves spatially local computations and applies to recurrent neural networks with static inputs. So far, hardware-oriented studies of Equilibrium Propagation focused on rate-based networks. In this work, we develop a spiking neural network algorithm called EqSpike, compatible with neuromorphic systems, which learns by Equilibrium Propagation. Through simulations, we obtain a test recognition accuracy of 96.9% on MNIST, similar to rate-based Equilibrium Propagation, and comparing favourably to alternative learning techniques for spiking neural networks. We show that EqSpike implemented in silicon neuromorphic technology could reduce the energy consumption of inference and training by up to three orders of magnitude compared to GPUs. Finally, we also show that during learning, EqSpike weight updates exhibit a form of Spike Timing Dependent Plasticity, highlighting a possible connection with biology.

Equilibrium Propagation with Continual Weight Updates

Equilibrium Propagation (EP) is a learning algorithm that bridges Machine Learning and Neuroscience, by computing gradients closely matching those of Backpropagation Through Time (BPTT), but with a learning rule local in space. Given an input $x$ and associated target $y$, EP proceeds in two phases: in the first phase neurons evolve freely towards a first steady state; in the second phase output neurons are nudged towards $y$ until they reach a second steady state. However, in existing implementations of EP, the learning rule is not local in time: the weight 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. In this work, we propose a version of EP named Continual Equilibrium Propagation (C-EP) where neuron and synapse dynamics occur simultaneously throughout the second phase, so that the weight update becomes local in time. Such a learning rule local both in space and time opens the possibility of an extremely energy efficient hardware implementation of EP. We prove theoretically that, provided the learning rates are sufficiently small, at each time step of the second phase the dynamics of neurons and synapses follow the gradients of the loss given by BPTT (Theorem 1). We demonstrate training with C-EP on MNIST and generalize C-EP to neural networks where neurons are connected by asymmetric connections. We show through experiments that the more the network updates follows the gradients of BPTT, the best it performs in terms of training. These results bring EP a step closer to biology by better complying with hardware constraints while maintaining its intimate link with backpropagation.

Continual Weight Updates and Convolutional Architectures for Equilibrium Propagation

Equilibrium Propagation (EP) is a biologically inspired alternative algorithm to backpropagation (BP) for training neural networks. It applies 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.

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

Equilibrium Propagation (EP) is a biologically inspired learning algorithm for convergent recurrent neural networks, i.e. RNNs that are fed 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.