Author name cluster

Rafal Bogacz

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15 papers

2 author rows

ICLR Conference 2025 Conference Paper

Benchmarking Predictive Coding Networks - Made Simple

Luca Pinchetti
Chang Qi
Oleh Lokshyn
Cornelius Emde
Amine M'Charrak
Mufeng Tang
Simon Frieder
Bayar Menzat

In this work, we tackle the problems of efficiency and scalability for predictive coding networks (PCNs) in machine learning. To do so, we propose a library that focuses on performance and simplicity, and use it to implement a large set of standard benchmarks for the community to use for their experiments. As most works in the field propose their own tasks and architectures, do not compare one against each other, and focus on small-scale tasks, a simple and fast open-source library, and a comprehensive set of benchmarks, would address all of these concerns. Then, we perform extensive tests on such benchmarks using both existing algorithms for PCNs, as well as adaptations of other methods popular in the bio-plausible deep learning community. All of this has allowed us to (i) test architectures much larger than commonly used in the literature, on more complex datasets; (ii) reach new state-of-the-art results in all of the tasks and dataset provided; (iii) clearly highlight what the current limitations of PCNs are, allowing us to state important future research directions. With the hope of galvanizing community efforts towards one of the main open problems in the field, scalability, we will release the code, tests, and benchmarks.

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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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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 2023 Conference Paper

Sequential Memory with Temporal Predictive Coding

Mufeng Tang
Helen Barron
Rafal Bogacz

Forming accurate memory of sequential stimuli is a fundamental function of biological agents. However, the computational mechanism underlying sequential memory in the brain remains unclear. Inspired by neuroscience theories and recent successes in applying predictive coding (PC) to \emph{static} memory tasks, in this work we propose a novel PC-based model for \emph{sequential} memory, called \emph{temporal predictive coding} (tPC). We show that our tPC models can memorize and retrieve sequential inputs accurately with a biologically plausible neural implementation. Importantly, our analytical study reveals that tPC can be viewed as a classical Asymmetric Hopfield Network (AHN) with an implicit statistical whitening process, which leads to more stable performance in sequential memory tasks of structured inputs. Moreover, we find that tPC exhibits properties consistent with behavioral observations and theories in neuroscience, thereby strengthening its biological relevance. Our work establishes a possible computational mechanism underlying sequential memory in the brain that can also be theoretically interpreted using existing memory model frameworks.

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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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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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AAAI Conference 2022 Conference Paper

Reverse Differentiation via Predictive Coding

Tommaso Salvatori
Yuhang Song
Zhenghua Xu
Thomas Lukasiewicz
Rafal Bogacz

Deep learning has redefined AI thanks to the rise of artificial neural networks, which are inspired by neuronal networks in the brain. Through the years, these interactions between AI and neuroscience have brought immense benefits to both fields, allowing neural networks to be used in a plethora of applications. Neural networks use an efficient implementation of reverse differentiation, called backpropagation (BP). This algorithm, however, is often criticized for its biological implausibility (e. g. , lack of local update rules for the parameters). Therefore, biologically plausible learning methods that rely on predictive coding (PC), a framework for describing information processing in the brain, are increasingly studied. Recent works prove that these methods can approximate BP up to a certain margin on multilayer perceptrons (MLPs), and asymptotically on any other complex model, and that zerodivergence inference learning (Z-IL), a variant of PC, is able to exactly implement BP on MLPs. However, the recent literature shows also that there is no biologically plausible method yet that can exactly replicate the weight update of BP on complex models. To fill this gap, in this paper, we generalize (PC and) Z-IL by directly defining it on computational graphs, and show that it can perform exact reverse differentiation. What results is the first PC (and so biologically plausible) algorithm that is equivalent to BP in the way of updating parameters on any neural network, providing a bridge between the interdisciplinary research of neuroscience and deep learning. Furthermore, the above results in particular also immediately provide a novel local and parallel implementation of BP.

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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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NeurIPS Conference 2021 Conference Paper

Associative Memories via Predictive Coding

Tommaso Salvatori
Yuhang Song
Yujian Hong
Lei Sha
Simon Frieder
Zhenghua Xu
Rafal Bogacz
Thomas Lukasiewicz

Associative memories in the brain receive and store patterns of activity registered by the sensory neurons, and are able to retrieve them when necessary. Due to their importance in human intelligence, computational models of associative memories have been developed for several decades now. In this paper, we present a novel neural model for realizing associative memories, which is based on a hierarchical generative network that receives external stimuli via sensory neurons. It is trained using predictive coding, an error-based learning algorithm inspired by information processing in the cortex. To test the model's capabilities, we perform multiple retrieval experiments from both corrupted and incomplete data points. In an extensive comparison, we show that this new model outperforms in retrieval accuracy and robustness popular associative memory models, such as autoencoders trained via backpropagation, and modern Hopfield networks. In particular, in completing partial data points, our model achieves remarkable results on natural image datasets, such as ImageNet, with a surprisingly high accuracy, even when only a tiny fraction of pixels of the original images is presented. Our model provides a plausible framework to study learning and retrieval of memories in the brain, as it closely mimics the behavior of the hippocampus as a memory index and generative model.

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NeurIPS Conference 2020 Conference Paper

Can the Brain Do Backpropagation? --- Exact Implementation of Backpropagation in Predictive Coding Networks

Yuhang Song
Thomas Lukasiewicz
Zhenghua Xu
Rafal Bogacz

Backpropagation (BP) has been the most successful algorithm used to train artificial neural networks. However, there are several gaps between BP and learning in biologically plausible neuronal networks of the brain (learning in the brain, or simply BL, for short), in particular, (1) it has been unclear to date, if BP can be implemented exactly via BL, (2) there is a lack of local plasticity in BP, i. e. , weight updates require information that is not locally available, while BL utilizes only locally available information, and (3)~there is a lack of autonomy in BP, i. e. , some external control over the neural network is required (e. g. , switching between prediction and learning stages requires changes to dynamics and synaptic plasticity rules), while BL works fully autonomously. Bridging such gaps, i. e. , understanding how BP can be approximated by BL, has been of major interest in both neuroscience and machine learning. Despite tremendous efforts, however, no previous model has bridged the gaps at a degree of demonstrating an equivalence to BP, instead, only approximations to BP have been shown. Here, we present for the first time a framework within BL that bridges the above crucial gaps. We propose a BL model that (1) produces \emph{exactly the same} updates of the neural weights as~BP, while (2)~employing local plasticity, i. e. , all neurons perform only local computations, done simultaneously. We then modify it to an alternative BL model that (3) also works fully autonomously. Overall, our work provides important evidence for the debate on the long-disputed question whether the brain can perform~BP.

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RLDM Conference 2013 Conference Abstract

Changing decision criteria in sequential decision making

Gaurav Malhotra
David Leslie
Rafal Bogacz

When making a sequence of decisions in a fixed amount of time, people need to decide how much evidence to accumulate before making each decision. Do people accumulate evidence to a fixed decision- criterion before making a decision or can this criterion change with time? What is the optimal shape of the decision criterion and how does this compare to the decision bounds adopted by people? We developed a theoretical model using average reward dynamic programming, that optimises the reward per unit time and shows that the optimal shape of the decision threshold is constant when all decisions in a sequence have a known difficulty. But in case of mixed difficulties, the model shows that optimal thresholds should change with time. We then conducted an experiment with human participants that parallels this model and found that decisions made by participants qualitatively matched this optimal model showing that participants infer the amount of evidence in perceptual stimuli and adjust their decision criterion based on this inference.

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YNIMG Journal 2010 Journal Article

The neural mechanisms of learning from competitors

Paul A. Howard-Jones
Rafal Bogacz
Jee H. Yoo
Ute Leonards
Skevi Demetriou

Learning from competitors poses a challenge for existing theories of reward-based learning, which assume that rewarded actions are more likely to be executed in the future. Such a learning mechanism would disadvantage a player in a competitive situation because, since the competitor's loss is the player's gain, reward might become associated with an action the player should themselves avoid. Using fMRI, we investigated the neural activity of humans competing with a computer in a foraging task. We observed neural activity that represented the variables required for learning from competitors: the actions of the competitor (in the player's motor and premotor cortex) and the reward prediction error arising from the competitor's feedback. In particular, regions positively correlated with the unexpected loss of the competitor (which was beneficial to the player) included the striatum and those regions previously implicated in response inhibition. Our results suggest that learning in such contexts may involve the competitor's unexpected losses activating regions of the player's brain that subserve response inhibition, as the player learns to avoid the actions that produced them.

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NeurIPS Conference 2000 Conference Paper

Emergence of Movement Sensitive Neurons' Properties by Learning a Sparse Code for Natural Moving Images

Rafal Bogacz
Malcolm Brown
Christophe Giraud-Carrier

Olshausen & Field demonstrated that a learning algorithm that attempts to generate a sparse code for natural scenes develops a complete family of localised, oriented, bandpass receptive fields, similar to those of 'simple cells' in VI. This paper describes an algorithm which finds a sparse code for sequences of images that preserves information about the input. This algorithm when trained on natural video sequences develops bases representing the movement in particular directions with particular speeds, similar to the receptive fields of the movement-sensitive cells observed in cortical visual areas. Furthermore, to previous approaches to learning direction selectivity, the timing of neuronal activity encodes the phase of the movement, so the precise timing of spikes is crucially important to the information encoding.

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