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Disentanglement is the endeavour to use machine learning to divide information about a dataset into meaningful fragments. In practice these fragments are representation (sub)spaces, often the set of channels in the latent space of a variational autoencoder (VAE). Assessments of disentanglement predominantly employ metrics that are coarse-grained at the model level, but this approach can obscure much about the process of information fragmentation. Here we propose to study the learned channels in aggregate, as the fragments of information learned by an ensemble of repeat training runs. Additionally, we depart from prior work where measures of similarity between individual subspaces neglected the nature of data embeddings as probability distributions. Instead, we view representation subspaces as communication channels that perform a soft clustering of the data; consequently, we generalize two classic information-theoretic measures of similarity between clustering assignments to compare representation spaces. We develop a lightweight method of estimation based on fingerprinting representation subspaces by their ability to distinguish dataset samples, allowing us to identify, analyze, and leverage meaningful structure in ensembles of VAEs trained on synthetic and natural datasets. Using this fully unsupervised pipeline we identify "hotspots" in the space of information fragments: groups of nearly identical representation subspaces that appear repeatedly in an ensemble of VAEs, particularly as regularization is increased. Finally, we leverage the proposed methodology to achieve ensemble learning with VAEs, boosting the information content of a set of weak learners -- a capability not possible with previous methods of assessing channel similarity., Comment: Code: https://github.com/murphyka/representation-space-info-comparison

Deterministic chaos permits a precise notion of a "perfect measurement" as one that, when obtained repeatedly, captures all of the information created by the system's evolution with minimal redundancy. Finding an optimal measurement is challenging, and has generally required intimate knowledge of the dynamics in the few cases where it has been done. We establish an equivalence between a perfect measurement and a variant of the information bottleneck. As a consequence, we can employ machine learning to optimize measurement processes that efficiently extract information from trajectory data. We obtain approximately optimal measurements for multiple chaotic maps and lay the necessary groundwork for efficient information extraction from general time series., Comment: Project page: https://distributed-information-bottleneck.github.io

Intrinsically motivated exploration has proven useful for reinforcement learning, even without additional extrinsic rewards. When the environment is naturally represented as a graph, how to guide exploration best remains an open question. In this work, we propose a novel approach for exploring graph-structured data motivated by two theories of human curiosity: the information gap theory and the compression progress theory. The theories view curiosity as an intrinsic motivation to optimize for topological features of subgraphs induced by nodes visited in the environment. We use these proposed features as rewards for graph neural-network-based reinforcement learning. On multiple classes of synthetically generated graphs, we find that trained agents generalize to longer exploratory walks and larger environments than are seen during training. Our method computes more efficiently than the greedy evaluation of the relevant topological properties. The proposed intrinsic motivations bear particular relevance for recommender systems. We demonstrate that next-node recommendations considering curiosity are more predictive of human choices than PageRank centrality in several real-world graph environments., Comment: 15 pages, 5 figures in main text, and 18 pages, 9 figures in supplement

One of the fundamental steps toward understanding a complex system is identifying variation at the scale of the system's components that is most relevant to behavior on a macroscopic scale. Mutual information provides a natural means of linking variation across scales of a system due to its independence of functional relationship between observables. However, characterizing the manner in which information is distributed across a set of observables is computationally challenging and generally infeasible beyond a handful of measurements. Here we propose a practical and general methodology that uses machine learning to decompose the information contained in a set of measurements by jointly optimizing a lossy compression of each measurement. Guided by the distributed information bottleneck as a learning objective, the information decomposition identifies the variation in the measurements of the system state most relevant to specified macroscale behavior. We focus our analysis on two paradigmatic complex systems: a Boolean circuit and an amorphous material undergoing plastic deformation. In both examples, the large amount of entropy of the system state is decomposed, bit by bit, in terms of what is most related to macroscale behavior. The identification of meaningful variation in data, with the full generality brought by information theory, is made practical for studying the connection between micro- and macroscale structure in complex systems., Comment: Project page: https://distributed-information-bottleneck.github.io

The model-free algorithms of “reinforcement learning” (RL) have gained clout across disciplines, but so too have model-based alternatives. The present study emphasizes other dimensions of this model space in consideration of associative or discriminative generalization across states and actions. This “generalized reinforcement learning” (GRL) model, a frugal extension of RL, parsimoniously retains the single reward-prediction error (RPE), but the scope of learning goes beyond the experienced state and action. Instead, the generalized RPE is efficiently relayed for bidirectional counterfactual updating of value estimates for other representations. Aided by structural information but as an implicit rather than explicit cognitive map, GRL provided the most precise account of human behavior and individual differences in a reversal-learning task with hierarchical structure that encouraged inverse generalization across both states and actions. Reflecting inference that could be true, false (i.e., overgeneralization), or absent (i.e., undergeneralization), state generalization distinguished those who learned well more so than action generalization. With high-resolution high-field fMRI targeting the dopaminergic midbrain, the GRL model's RPE signals (alongside value and decision signals) were localized within not only the striatum but also the substantia nigra and the ventral tegmental area, including specific effects of generalization that also extend to the hippocampus. Factoring in generalization as a multidimensional process in value-based learning, these findings shed light on complexities that, while challenging classic RL, can still be resolved within the bounds of its core computations.

Interpretability is a pressing issue for machine learning. Common approaches to interpretable machine learning constrain interactions between features of the input, rendering the effects of those features on a model's output comprehensible but at the expense of model complexity. We approach interpretability from a new angle: constrain the information about the features without restricting the complexity of the model. Borrowing from information theory, we use the Distributed Information Bottleneck to find optimal compressions of each feature that maximally preserve information about the output. The learned information allocation, by feature and by feature value, provides rich opportunities for interpretation, particularly in problems with many features and complex feature interactions. The central object of analysis is not a single trained model, but rather a spectrum of models serving as approximations that leverage variable amounts of information about the inputs. Information is allocated to features by their relevance to the output, thereby solving the problem of feature selection by constructing a learned continuum of feature inclusion-to-exclusion. The optimal compression of each feature -- at every stage of approximation -- allows fine-grained inspection of the distinctions among feature values that are most impactful for prediction. We develop a framework for extracting insight from the spectrum of approximate models and demonstrate its utility on a range of tabular datasets., Comment: project page: https://distributed-information-bottleneck.github.io

A hallmark of chaotic dynamics is the loss of information with time. Although information loss is often expressed through a connection to Lyapunov exponents -- valid in the limit of high information about the system state -- this picture misses the rich spectrum of information decay across different levels of granularity. Here we show how machine learning presents new opportunities for the study of information loss in chaotic dynamics, with a double pendulum serving as a model system. We use the Information Bottleneck as a training objective for a neural network to extract information from the state of the system that is optimally predictive of the future state after a prescribed time horizon. We then decompose the optimally predictive information by distributing a bottleneck to each state variable, recovering the relative importance of the variables in determining future evolution. The framework we develop is broadly applicable to chaotic systems and pragmatic to apply, leveraging data and machine learning to monitor the limits of predictability and map out the loss of information., Comment: NeurIPS 2022 workshop paper (Machine learning and the physical sciences); project page: distributed-information-bottleneck.github.io

The structure of knowledge is commonly described as a network of key concepts and semantic relations between them. A learner of a particular domain can discover this network by navigating the nodes and edges presented by instructional material, such as a textbook, workbook, or other text. While over a long temporal period such exploration processes are certain to discover the whole connected network, little is known about how the learning is affected by the dual pressures of finite study time and human mental errors. Here we model the learning of linear algebra textbooks with finite length random walks over the corresponding semantic networks. We show that if a learner does not keep up with the pace of material presentation, the learning can be an order of magnitude worse than it is in the asymptotic limit. Further, we find that this loss is compounded by three types of mental errors: forgetting, shuffling, and reinforcement. Broadly, our study informs the design of teaching materials from both structural and temporal perspectives., Comment: 29 RevTeX pages, 13 figures

In multiple academic disciplines, having a perceived gender of `woman' is associated with a lower than expected rate of citations. In some fields, that disparity is driven primarily by the citations of men and is increasing over time despite increasing diversification of the profession. It is likely that complex social interactions and individual ideologies shape these disparities. Computational models of select factors that reproduce empirical observations can help us understand some of the minimal driving forces behind these complex phenomena and therefore aid in their mitigation. Here, we present a simple agent-based model of citation practices within academia, in which academics generate citations based on three factors: their estimate of the collaborative network of the field, how they sample that estimate, and how open they are to learning about their field from other academics. We show that increasing homophily -- or the tendency of people to interact with others more like themselves -- in these three domains is sufficient to reproduce observed biases in citation practices. We find that homophily in sampling an estimate of the field influences total citation rates, and openness to learning from new and unfamiliar authors influences the change in those citations over time. We next model a real-world intervention -- the citation diversity statement -- which has the potential to influence both of these parameters. We determine a parameterization of our model that matches the citation practices of academics who use the citation diversity statement. This parameterization paired with an openness to learning from many new authors can result in citation practices that are equitable and stable over time. Ultimately, our work underscores the importance of homophily in shaping citation practices and provides evidence that specific actions may mitigate biased citation practices in academia.

The fruits of science are relationships made comprehensible, often by way of approximation. While deep learning is an extremely powerful way to find relationships in data, its use in science has been hindered by the difficulty of understanding the learned relationships. The Information Bottleneck (IB) is an information theoretic framework for understanding a relationship between an input and an output in terms of a trade-off between the fidelity and complexity of approximations to the relationship. Here we show that a crucial modification -- distributing bottlenecks across multiple components of the input -- opens fundamentally new avenues for interpretable deep learning in science. The Distributed Information Bottleneck throttles the downstream complexity of interactions between the components of the input, deconstructing a relationship into meaningful approximations found through deep learning without requiring custom-made datasets or neural network architectures. Applied to a complex system, the approximations illuminate aspects of the system's nature by restricting -- and monitoring -- the information about different components incorporated into the approximation. We demonstrate the Distributed IB's explanatory utility in systems drawn from applied mathematics and condensed matter physics. In the former, we deconstruct a Boolean circuit into approximations that isolate the most informative subsets of input components without requiring exhaustive search. In the latter, we localize information about future plastic rearrangement in the static structure of a sheared glass, and find the information to be more or less diffuse depending on the system's preparation. By way of a principled scheme of approximations, the Distributed IB brings much-needed interpretability to deep learning and enables unprecedented analysis of information flow through a system.

Random walks are a common model for exploration and discovery of complex networks. While numerous algorithms have been proposed to map out an unknown network, a complementary question arises: in a known network, which nodes and edges are most likely to be discovered by a random walker in finite time? Here we introduce exposure theory, a statistical mechanics framework that predicts the learning of nodes and edges across several types of networks, including weighted and temporal, and show that edge learning follows a universal trajectory. While the learning of individual nodes and edges is noisy, exposure theory produces a highly accurate prediction of aggregate exploration statistics., Comment: 15 RevTeX pages, 8 figures

Network control theory is increasingly used to profile the brain's energy landscape via simulations of neural dynamics. This approach estimates the control energy required to simulate the activation of brain circuits based on structural connectome measured using diffusion magnetic resonance imaging, thereby quantifying those circuits' energetic efficiency. The biological basis of control energy, however, remains unknown, hampering its further application. To fill this gap, investigating temporal lobe epilepsy as a lesion model, we show that patients require higher control energy to activate the limbic network than healthy volunteers, especially ipsilateral to the seizure focus. The energetic imbalance between ipsilateral and contralateral temporolimbic regions is tracked by asymmetric patterns of glucose metabolism measured using positron emission tomography, which, in turn, may be selectively explained by asymmetric gray matter loss as evidenced in the hippocampus. Our investigation provides the first theoretical framework unifying gray matter integrity, metabolism, and energetic generation of neural dynamics.

The historical and contemporary under-attribution of women's contributions to scientific scholarship is well-known and well-studied, with effects that are felt today in myriad ways by women scientists. One measure of this under-attribution is the so-called citation gap between men and women: the under-citation of papers authored by women relative to expected rates coupled with a corresponding over-citation of papers authored by men relative to expected rates. We explore the citation gap in contemporary physics, analyzing over one million articles published over the last 25 years in 35 physics journals that span a wide range of subfields. Using a model that predicts papers' expected citation rates according to a set of characteristics separate from author gender, we find a global bias wherein papers authored by women are significantly under-cited, and papers authored by men are significantly over-cited. Moreover, we find that citation behavior varies along several dimensions, such that imbalances differ according to who is citing, where they are citing, and how they are citing. Specifically, citation imbalance in favor of man-authored papers is highest for papers authored by men, papers published in general physics journals, and papers likely to be less familiar to citing authors. Our results suggest that, although deciding which papers to cite is an individual choice, the cumulative effects of these choices needlessly harm a subset of scholars. We discuss several strategies for the mitigation of these effects, including conscious behavioral changes at the individual, journal, and community levels.

Neural systems are well known for their ability to learn and store information as memories. Even more impressive is their ability to abstract these memories to create complex internal representations, enabling advanced functions such as the spatial manipulation of mental representations. While recurrent neural networks (RNNs) are capable of representing complex information, the exact mechanisms of how dynamical neural systems perform abstraction are still not well-understood, thereby hindering the development of more advanced functions. Here, we train a 1000-neuron RNN -- a reservoir computer (RC) -- to abstract a continuous dynamical attractor memory from isolated examples of dynamical attractor memories. Further, we explain the abstraction mechanism with new theory. By training the RC on isolated and shifted examples of either stable limit cycles or chaotic Lorenz attractors, the RC learns a continuum of attractors, as quantified by an extra Lyapunov exponent equal to zero. We propose a theoretical mechanism of this abstraction by combining ideas from differentiable generalized synchronization and feedback dynamics. Our results quantify abstraction in simple neural systems, enabling us to design artificial RNNs for abstraction, and leading us towards a neural basis of abstraction., Comment: 9 pages