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Beren Millidge

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10 papers

2 author rows

ICLR Conference 2024 Conference Paper

A Stable, Fast, and Fully Automatic Learning Algorithm for Predictive Coding Networks

Tommaso Salvatori
Yuhang Song 0001
Yordan Yordanov
Beren Millidge
Lei Sha
Cornelius Emde
Zhenghua Xu 0001
Rafal Bogacz

Predictive coding networks are neuroscience-inspired models with roots in both Bayesian statistics and neuroscience. Training such models, however, is quite inefficient and unstable. In this work, we show how by simply changing the temporal scheduling of the update rule for the synaptic weights leads to an algorithm that is much more efficient and stable than the original one, and has theoretical guarantees in terms of convergence. The proposed algorithm, that we call incremental predictive coding (iPC) is also more biologically plausible than the original one, as it it fully automatic. In an extensive set of experiments, we show that iPC constantly performs better than the original formulation on a large number of benchmarks for image classification, as well as for the training of both conditional and masked language models, in terms of test accuracy, efficiency, and convergence with respect to a large set of hyperparameters.

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ICML Conference 2024 Conference Paper

Predictive Coding beyond Correlations

Tommaso Salvatori
Luca Pinchetti
Amine M'Charrak
Beren Millidge
Thomas Lukasiewicz

Biologically plausible learning algorithms offer a promising alternative to traditional deep learning techniques, especially in overcoming the limitations of backpropagation in fast and low-energy neuromorphic implementations. To this end, there has been extensive research in understanding what their capabilities are. In this work, we show how one of such algorithms, called predictive coding, is able to perform causal inference tasks. First, we show how a simple change in the inference process of predictive coding enables to compute interventions without the need to mutilate or redefine a causal graph. Then, we explore applications in cases where the graph is unknown, and has to be inferred from observational data. Empirically, we show how such findings can be used to improve the performance of predictive coding in image classification tasks, and conclude that such models are naturally able to perform causal inference tasks using a biologically plausible kind of message passing.

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ICLR Conference 2023 Conference Paper

A Theoretical Framework for Inference and Learning in Predictive Coding Networks

Beren Millidge
Yuhang Song 0001
Tommaso Salvatori
Thomas Lukasiewicz
Rafal Bogacz

Predictive coding (PC) is an influential theory in computational neuroscience, which argues that the cortex forms unsupervised world models by implementing a hierarchical process of prediction error minimization. PC networks (PCNs) are trained in two phases. First, neural activities are updated to optimize the network's response to external stimuli. Second, synaptic weights are updated to consolidate this change in activity --- an algorithm called \emph{prospective configuration}. While previous work has shown how in various limits, PCNs can be found to approximate backpropagation (BP), recent work has demonstrated that PCNs operating in this standard regime, which does not approximate BP, nevertheless obtain competitive training and generalization performance to BP-trained networks while outperforming them on various tasks. However, little is understood theoretically about the properties and dynamics of PCNs in this regime. In this paper, we provide a comprehensive theoretical analysis of the properties of PCNs trained with prospective configuration. We first derive analytical results concerning the inference equilibrium for PCNs and a previously unknown close connection relationship to target propagation (TP). Secondly, we provide a theoretical analysis of learning in PCNs as a variant of generalized expectation-maximization and use that to prove the convergence of PCNs to critical points of the BP loss function, thus showing that deep PCNs can, in theory, achieve the same generalization performance as BP, while maintaining their unique advantages.

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ECAI Conference 2023 Conference Paper

Associative Memories in the Feature Space

Tommaso Salvatori
Beren Millidge
Yuhang Song 0001
Rafal Bogacz
Thomas Lukasiewicz

An autoassociative memory model is a function that, given a set of data points, takes as input an arbitrary vector and outputs the most similar data point from the memorized set. However, popular memory models fail to retrieve images even when the corruption is mild and easy to detect for a human evaluator. This is because similarities are evaluated in the raw pixel space, which does not contain any semantic information about the images. This problem can be easily solved by computing similarities in an embedding space instead of the pixel space. We show that an effective way of computing such embeddings is via a network pretrained with a contrastive loss. As the dimension of embedding spaces is often significantly smaller than the pixel space, we also have a faster computation of similarity scores. We test this method on complex datasets such as CIFAR10 and STL10. An additional drawback of current models is the need of storing the whole dataset in the pixel space, which is often extremely large. We relax this condition and propose a class of memory models that only stores low-dimensional semantic embeddings, and uses them to retrieve similar, but not identical, memories. We demonstrate a proof of concept of this method on a simple task on the MNIST dataset.

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ICLR Conference 2023 Conference Paper

Backpropagation at the Infinitesimal Inference Limit of Energy-Based Models: Unifying Predictive Coding, Equilibrium Propagation, and Contrastive Hebbian Learning

Beren Millidge
Yuhang Song 0001
Tommaso Salvatori
Thomas Lukasiewicz
Rafal Bogacz

How the brain performs credit assignment is a fundamental unsolved problem in neuroscience. Many `biologically plausible' algorithms have been proposed, which compute gradients that approximate those computed by backpropagation (BP), and which operate in ways that more closely satisfy the constraints imposed by neural circuitry. Many such algorithms utilize the framework of energy-based models (EBMs), in which all free variables in the model are optimized to minimize a global energy function. However, in the literature, these algorithms exist in isolation and no unified theory exists linking them together. Here, we provide a comprehensive theory of the conditions under which EBMs can approximate BP, which lets us unify many of the BP approximation results in the literature (namely, predictive coding, equilibrium propagation, and contrastive Hebbian learning) and demonstrate that their approximation to BP arises from a simple and general mathematical property of EBMs at free-phase equilibrium. This property can then be exploited in different ways with different energy functions, and these specific choices yield a family of BP-approximating algorithms, which both includes the known results in the literature and can be used to derive new ones.

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NeurIPS Conference 2022 Conference Paper

A Theoretical Framework for Inference Learning

Nick Alonso
Beren Millidge
Jeffrey Krichmar
Emre O Neftci

Backpropagation (BP) is the most successful and widely used algorithm in deep learning. However, the computations required by BP are challenging to reconcile with known neurobiology. This difficulty has stimulated interest in more biologically plausible alternatives to BP. One such algorithm is the inference learning algorithm (IL). IL trains predictive coding models of neural circuits and has achieved equal performance to BP on supervised and auto-associative tasks. In contrast to BP, however, the mathematical foundations of IL are not well-understood. Here, we develop a novel theoretical framework for IL. Our main result is that IL closely approximates an optimization method known as implicit stochastic gradient descent (implicit SGD), which is distinct from the explicit SGD implemented by BP. Our results further show how the standard implementation of IL can be altered to better approximate implicit SGD. Our novel implementation considerably improves the stability of IL across learning rates, which is consistent with our theory, as a key property of implicit SGD is its stability. We provide extensive simulation results that further support our theoretical interpretations and find IL achieves quicker convergence when trained with mini-batch size one while performing competitively with BP for larger mini-batches when combined with Adam.

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NeurIPS Conference 2022 Conference Paper

Learning on Arbitrary Graph Topologies via Predictive Coding

Tommaso Salvatori
Luca Pinchetti
Beren Millidge
Yuhang Song
Tianyi Bao
Rafal Bogacz
Thomas Lukasiewicz

Training with backpropagation (BP) in standard deep learning consists of two main steps: a forward pass that maps a data point to its prediction, and a backward pass that propagates the error of this prediction back through the network. This process is highly effective when the goal is to minimize a specific objective function. However, it does not allow training on networks with cyclic or backward connections. This is an obstacle to reaching brain-like capabilities, as the highly complex heterarchical structure of the neural connections in the neocortex are potentially fundamental for its effectiveness. In this paper, we show how predictive coding (PC), a theory of information processing in the cortex, can be used to perform inference and learning on arbitrary graph topologies. We experimentally show how this formulation, called PC graphs, can be used to flexibly perform different tasks with the same network by simply stimulating specific neurons. This enables the model to be queried on stimuli with different structures, such as partial images, images with labels, or images without labels. We conclude by investigating how the topology of the graph influences the final performance, and comparing against simple baselines trained with BP.

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NeurIPS Conference 2022 Conference Paper

Predictive Coding beyond Gaussian Distributions

Luca Pinchetti
Tommaso Salvatori
Yordan Yordanov
Beren Millidge
Yuhang Song
Thomas Lukasiewicz

A large amount of recent research has the far-reaching goal of finding training methods for deep neural networks that can serve as alternatives to backpropagation~(BP). A prominent example is predictive coding (PC), which is a neuroscience-inspired method that performs inference on hierarchical Gaussian generative models. These methods, however, fail to keep up with modern neural networks, as they are unable to replicate the dynamics of complex layers and activation functions. In this work, we solve this problem by generalizing PC to arbitrary probability distributions, enabling the training of architectures, such as transformers, that are hard to approximate with only Gaussian assumptions. We perform three experimental analyses. First, we study the gap between our method and the standard formulation of PC on multiple toy examples. Second, we test the reconstruction quality on variational autoencoders, where our method reaches the same reconstruction quality as BP. Third, we show that our method allows us to train transformer networks and achieve performance comparable with BP on conditional language models. More broadly, this method allows neuroscience-inspired learning to be applied to multiple domains, since the internal distributions can be flexibly adapted to the data, tasks, and architectures used.

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IJCAI Conference 2022 Conference Paper

Predictive Coding: Towards a Future of Deep Learning beyond Backpropagation?

Beren Millidge
Tommaso Salvatori
Yuhang Song
Rafal Bogacz
Thomas Lukasiewicz

The backpropagation of error algorithm (BP) used to train deep neural networks has been fundamental to the successes of deep learning. However, it requires sequential backwards updates and non-local computations which make it challenging to parallelize at scale and is unlike how learning works in the brain. Neuroscience-inspired learning algorithms, however, such as \emph{predictive coding} which utilize local learning have the potential to overcome these limitations and advance beyond deep learning technologies in the future. While predictive coding originated in theoretical neuroscience as a model of information processing in the cortex, recent work has developed the idea into a general-purpose algorithm able to train neural networks using only local computations. In this survey, we review works that have contributed to this perspective and demonstrate the close connection between predictive coding and backpropagation in terms of generalization quality, as well as works that highlight the multiple advantages of using predictive coding models over backprop-trained neural networks. Specifically, we show the substantially greater flexibility of predictive coding networks against equivalent deep neural networks, which can function as classifiers, generators, and associative memories simultaneously, and can be defined on arbitrary graph topologies. Finally, we review direct benchmarks of predictive coding networks on machine learning classification tasks, as well as its close connections to control theory and applications in robotics.

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ICML Conference 2022 Conference Paper

Universal Hopfield Networks: A General Framework for Single-Shot Associative Memory Models

Beren Millidge
Tommaso Salvatori
Yuhang Song 0001
Thomas Lukasiewicz
Rafal Bogacz

A large number of neural network models of associative memory have been proposed in the literature. These include the classical Hopfield networks (HNs), sparse distributed memories (SDMs), and more recently the modern continuous Hopfield networks (MCHNs), which possess close links with self-attention in machine learning. In this paper, we propose a general framework for understanding the operation of such memory networks as a sequence of three operations: similarity, separation, and projection. We derive all these memory models as instances of our general framework with differing similarity and separation functions. We extend the mathematical framework of Krotov et al (2020) to express general associative memory models using neural network dynamics with local computation, and derive a general energy function that is a Lyapunov function of the dynamics. Finally, using our framework, we empirically investigate the capacity of using different similarity functions for these associative memory models, beyond the dot product similarity measure, and demonstrate empirically that Euclidean or Manhattan distance similarity metrics perform substantially better in practice on many tasks, enabling a more robust retrieval and higher memory capacity than existing models.

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