arXiv subjects

Beren Millidge

Publications and source records attributed to Beren Millidge.

At least 55 records · Page 3Linked to original sources

arXiv2022

Capsule Networks as Generative Models

Capsule networks are a neural network architecture specialized for visual scene recognition. Features and pose information are extracted from a scene and then dynamically routed through a hierarchy of vector-valued nodes called 'capsules' to create an implicit scene graph, with the ultimate aim of learning vision directly as inverse graphics. Despite these intuitions, however, capsule networks are not formulated as explicit probabilistic generative models; moreover, the routing algorithms typically used are ad-hoc and primarily motivated by algorithmic intuition. In this paper, we derive an alternative capsule routing algorithm utilizing iterative inference under sparsity constraints. We then introduce an explicit probabilistic generative model for capsule networks based on the self-attention operation in transformer networks and show how it is related to a variant of predictive coding networks using Von-Mises-Fisher (VMF) circular Gaussian distributions.

q-bio.NC↗

arXiv2022

Preventing Deterioration of Classification Accuracy in Predictive Coding Networks

Predictive Coding Networks (PCNs) aim to learn a generative model of the world. Given observations, this generative model can then be inverted to infer the causes of those observations. However, when training PCNs, a noticeable pathology is often observed where inference accuracy peaks and then declines with further training. This cannot be explained by overfitting since both training and test accuracy decrease simultaneously. Here we provide a thorough investigation of this phenomenon and show that it is caused by an imbalance between the speeds at which the various layers of the PCN converge. We demonstrate that this can be prevented by regularising the weight matrices at each layer: by restricting the relative size of matrix singular values, we allow the weight matrix to change but restrict the overall impact which a layer can have on its neighbours. We also demonstrate that a similar effect can be achieved through a more biologically plausible and simple scheme of just capping the weights.

cs.AI↗

arXiv2022

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

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.

cs.LG↗

arXiv2022

A Theoretical Framework for Inference and Learning in Predictive Coding Networks

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 tasks such as online, few-shot, and continual learning, where brains are known to excel. Despite this promising empirical performance, 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.

cs.NE↗

arXiv2022

Successor Representation Active Inference

Recent work has uncovered close links between between classical reinforcement learning algorithms, Bayesian filtering, and Active Inference which lets us understand value functions in terms of Bayesian posteriors. An alternative, but less explored, model-free RL algorithm is the successor representation, which expresses the value function in terms of a successor matrix of expected future state occupancies. In this paper, we derive the probabilistic interpretation of the successor representation in terms of Bayesian filtering and thus design a novel active inference agent architecture utilizing successor representations instead of model-based planning. We demonstrate that active inference successor representations have significant advantages over current active inference agents in terms of planning horizon and computational cost. Moreover, we demonstrate how the successor representation agent can generalize to changing reward functions such as variants of the expected free energy.

cs.AI↗

arXiv2022

Predictive Coding: a Theoretical and Experimental Review

Predictive coding offers a potentially unifying account of cortical function -- postulating that the core function of the brain is to minimize prediction errors with respect to a generative model of the world. The theory is closely related to the Bayesian brain framework and, over the last two decades, has gained substantial influence in the fields of theoretical and cognitive neuroscience. A large body of research has arisen based on both empirically testing improved and extended theoretical and mathematical models of predictive coding, as well as in evaluating their potential biological plausibility for implementation in the brain and the concrete neurophysiological and psychological predictions made by the theory. Despite this enduring popularity, however, no comprehensive review of predictive coding theory, and especially of recent developments in this field, exists. Here, we provide a comprehensive review both of the core mathematical structure and logic of predictive coding, thus complementing recent tutorials in the literature. We also review a wide range of classic and recent work within the framework, ranging from the neurobiologically realistic microcircuits that could implement predictive coding, to the close relationship between predictive coding and the widely-used backpropagation of error algorithm, as well as surveying the close relationships between predictive coding and modern machine learning techniques.

cs.AI↗

arXiv2022

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

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 possesses 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 only second-order interactions between neurons, 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.

cs.NE↗

arXiv2022

A Theoretical Framework for Inference Learning

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 has close connections to neurobiological models of cortical function and has achieved equal performance to BP on supervised learning 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 also demonstrate IL achieves quicker convergence when trained with small mini-batches while matching the performance of BP for large mini-batches.

cs.NE↗

arXiv2022

How particular is the physics of the free energy principle?

The free energy principle (FEP) states that any dynamical system can be interpreted as performing Bayesian inference upon its surrounding environment. In this work, we examine in depth the assumptions required to derive the FEP in the simplest possible set of systems -- weakly-coupled non-equilibrium linear stochastic systems. Specifically, we explore (i) how general the requirements imposed on the statistical structure of a system are and (ii) how informative the FEP is about the behaviour of such systems. We discover that two requirements of the FEP -- the Markov blanket condition (i.e. a statistical boundary precluding direct coupling between internal and external states) and stringent restrictions on its solenoidal flows (i.e. tendencies driving a system out of equilibrium) -- are only valid for a very narrow space of parameters. Suitable systems require an absence of perception-action asymmetries that is highly unusual for living systems interacting with an environment. More importantly, we observe that a mathematically central step in the argument, connecting the behaviour of a system to variational inference, relies on an implicit equivalence between the dynamics of the average states of a system with the average of the dynamics of those states. This equivalence does not hold in general even for linear systems, since it requires an effective decoupling from the system's history of interactions. These observations are critical for evaluating the generality and applicability of the FEP and indicate the existence of significant problems of the theory in its current form. These issues make the FEP, as it stands, not straightforwardly applicable to the simple linear systems studied here and suggest that more development is needed before the theory could be applied to the kind of complex systems that describe living and cognitive processes.

q-bio.NC↗

arXiv2022

pymdp: A Python library for active inference in discrete state spaces

Active inference is an account of cognition and behavior in complex systems which brings together action, perception, and learning under the theoretical mantle of Bayesian inference. Active inference has seen growing applications in academic research, especially in fields that seek to model human or animal behavior. While in recent years, some of the code arising from the active inference literature has been written in open source languages like Python and Julia, to-date, the most popular software for simulating active inference agents is the DEM toolbox of SPM, a MATLAB library originally developed for the statistical analysis and modelling of neuroimaging data. Increasing interest in active inference, manifested both in terms of sheer number as well as diversifying applications across scientific disciplines, has thus created a need for generic, widely-available, and user-friendly code for simulating active inference in open-source scientific computing languages like Python. The Python package we present here, pymdp (see https://github.com/infer-actively/pymdp), represents a significant step in this direction: namely, we provide the first open-source package for simulating active inference with partially-observable Markov Decision Processes or POMDPs. We review the package's structure and explain its advantages like modular design and customizability, while providing in-text code blocks along the way to demonstrate how it can be used to build and run active inference processes with ease. We developed pymdp to increase the accessibility and exposure of the active inference framework to researchers, engineers, and developers with diverse disciplinary backgrounds. In the spirit of open-source software, we also hope that it spurs new innovation, development, and collaboration in the growing active inference community.

cs.AI↗

arXiv2022

Hybrid Predictive Coding: Inferring, Fast and Slow

Predictive coding is an influential model of cortical neural activity. It proposes that perceptual beliefs are furnished by sequentially minimising "prediction errors" - the differences between predicted and observed data. Implicit in this proposal is the idea that perception requires multiple cycles of neural activity. This is at odds with evidence that several aspects of visual perception - including complex forms of object recognition - arise from an initial "feedforward sweep" that occurs on fast timescales which preclude substantial recurrent activity. Here, we propose that the feedforward sweep can be understood as performing amortized inference and recurrent processing can be understood as performing iterative inference. We propose a hybrid predictive coding network that combines both iterative and amortized inference in a principled manner by describing both in terms of a dual optimization of a single objective function. We show that the resulting scheme can be implemented in a biologically plausible neural architecture that approximates Bayesian inference utilising local Hebbian update rules. We demonstrate that our hybrid predictive coding model combines the benefits of both amortized and iterative inference -- obtaining rapid and computationally cheap perceptual inference for familiar data while maintaining the context-sensitivity, precision, and sample efficiency of iterative inference schemes. Moreover, we show how our model is inherently sensitive to its uncertainty and adaptively balances iterative and amortized inference to obtain accurate beliefs using minimum computational expense. Hybrid predictive coding offers a new perspective on the functional relevance of the feedforward and recurrent activity observed during visual perception and offers novel insights into distinct aspects of visual phenomenology.

q-bio.NC↗

arXiv2022

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

The backpropagation of error algorithm used to train deep neural networks has been fundamental to the successes of deep learning. However, it requires sequential backward 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 current deep learning technologies. 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 theoretical connections between predictive coding and backpropagation, as well as works that highlight the multiple advantages of using predictive coding models over backpropagation-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.

cs.NE↗

arXiv2021

Active Inference in Robotics and Artificial Agents: Survey and Challenges

Active inference is a mathematical framework which originated in computational neuroscience as a theory of how the brain implements action, perception and learning. Recently, it has been shown to be a promising approach to the problems of state-estimation and control under uncertainty, as well as a foundation for the construction of goal-driven behaviours in robotics and artificial agents in general. Here, we review the state-of-the-art theory and implementations of active inference for state-estimation, control, planning and learning; describing current achievements with a particular focus on robotics. We showcase relevant experiments that illustrate its potential in terms of adaptation, generalization and robustness. Furthermore, we connect this approach with other frameworks and discuss its expected benefits and challenges: a unified framework with functional biological plausibility using variational Bayesian inference.

cs.RO↗

arXiv2021

Understanding the Origin of Information-Seeking Exploration in Probabilistic Objectives for Control

The exploration-exploitation trade-off is central to the description of adaptive behaviour in fields ranging from machine learning, to biology, to economics. While many approaches have been taken, one approach to solving this trade-off has been to equip or propose that agents possess an intrinsic 'exploratory drive' which is often implemented in terms of maximizing the agents information gain about the world -- an approach which has been widely studied in machine learning and cognitive science. In this paper we mathematically investigate the nature and meaning of such approaches and demonstrate that this combination of utility maximizing and information-seeking behaviour arises from the minimization of an entirely difference class of objectives we call divergence objectives. We propose a dichotomy in the objective functions underlying adaptive behaviour between \emph{evidence} objectives, which correspond to well-known reward or utility maximizing objectives in the literature, and \emph{divergence} objectives which instead seek to minimize the divergence between the agent's expected and desired futures, and argue that this new class of divergence objectives could form the mathematical foundation for a much richer understanding of the exploratory components of adaptive and intelligent action, beyond simply greedy utility maximization.

cs.LG↗

arXiv2021

A Mathematical Walkthrough and Discussion of the Free Energy Principle

The Free-Energy-Principle (FEP) is an influential and controversial theory which postulates a deep and powerful connection between the stochastic thermodynamics of self-organization and learning through variational inference. Specifically, it claims that any self-organizing system which can be statistically separated from its environment, and which maintains itself at a non-equilibrium steady state, can be construed as minimizing an information-theoretic functional -- the variational free energy -- and thus performing variational Bayesian inference to infer the hidden state of its environment. This principle has also been applied extensively in neuroscience, and is beginning to make inroads in machine learning by spurring the construction of novel and powerful algorithms by which action, perception, and learning can all be unified under a single objective. While its expansive and often grandiose claims have spurred significant debates in both philosophy and theoretical neuroscience, the mathematical depth and lack of accessible introductions and tutorials for the core claims of the theory have often precluded a deep understanding within the literature. Here, we aim to provide a mathematically detailed, yet intuitive walk-through of the formulation and central claims of the FEP while also providing a discussion of the assumptions necessary and potential limitations of the theory. Additionally, since the FEP is a still a living theory, subject to internal controversy, change, and revision, we also present a detailed appendix highlighting and condensing current perspectives as well as controversies about the nature, applicability, and the mathematical assumptions and formalisms underlying the FEP.

cs.AI↗

arXiv2021

Habitual and Reflective Control in Hierarchical Predictive Coding

In cognitive science, behaviour is often separated into two types. Reflexive control is habitual and immediate, whereas reflective is deliberative and time consuming. We examine the argument that Hierarchical Predictive Coding (HPC) can explain both types of behaviour as a continuum operating across a multi-layered network, removing the need for separate circuits in the brain. On this view, "fast" actions may be triggered using only the lower layers of the HPC schema, whereas more deliberative actions need higher layers. We demonstrate that HPC can distribute learning throughout its hierarchy, with higher layers called into use only as required.

cs.AI↗

arXiv2021

Applications of the Free Energy Principle to Machine Learning and Neuroscience

In this PhD thesis, we explore and apply methods inspired by the free energy principle to two important areas in machine learning and neuroscience. The free energy principle is a general mathematical theory of the necessary information-theoretic behaviours of systems that maintain a separation from their environment. A core postulate of the theory is that complex systems can be seen as performing variational Bayesian inference and minimizing an information-theoretic quantity called the variational free energy. The thesis is structured into three independent sections. Firstly, we focus on predictive coding, a neurobiologically plausible process theory derived from the free energy principle which argues that the primary function of the brain is to minimize prediction errors, showing how predictive coding can be scaled up and extended to be more biologically plausible, and elucidating its close links with other methods such as Kalman Filtering. Secondly, we study active inference, a neurobiologically grounded account of action through variational message passing, and investigate how these methods can be scaled up to match the performance of deep reinforcement learning methods. We additionally provide a detailed mathematical understanding of the nature and origin of the information-theoretic objectives that underlie exploratory behaviour. Finally, we investigate biologically plausible methods of credit assignment in the brain. We first demonstrate a close link between predictive coding and the backpropagation of error algorithm. We go on to propose novel and simpler algorithms which allow for backprop to be implemented in purely local, biologically plausible computations.

cs.AI↗

arXiv2021

Towards a Mathematical Theory of Abstraction

While the utility of well-chosen abstractions for understanding and predicting the behaviour of complex systems is well appreciated, precisely what an abstraction $\textit{is}$ has so far has largely eluded mathematical formalization. In this paper, we aim to set out a mathematical theory of abstraction. We provide a precise characterisation of what an abstraction is and, perhaps more importantly, suggest how abstractions can be learnt directly from data both for static datasets and for dynamical systems. We define an abstraction to be a small set of `summaries' of a system which can be used to answer a set of queries about the system or its behaviour. The difference between the ground truth behaviour of the system on the queries and the behaviour of the system predicted only by the abstraction provides a measure of the `leakiness' of the abstraction which can be used as a loss function to directly learn abstractions from data. Our approach can be considered a generalization of classical statistics where we are not interested in reconstructing `the data' in full, but are instead only concerned with answering a set of arbitrary queries about the data. While highly theoretical, our results have deep implications for statistical inference and machine learning and could be used to develop explicit methods for learning precise kinds of abstractions directly from data.

cs.AI↗