Exact and Soft Successive Refinement of the Information Bottleneck.

The information bottleneck (IB) framework formalises the essential requirement for efficient information processing systems to achieve an optimal balance between the complexity of their representation and the amount of information extracted about relevant features. However, since the representation complexity affordable by real-world systems may vary in time, the processing cost of updating the representations should also be taken into account. A crucial question is thus the extent to which adaptive systems can leverage the information content of already existing IB-optimal representations for producing new ones, which target the same relevant features but at a different granularity. We investigate the information-theoretic optimal limits of this process by studying and extending, within the IB framework, the notion of successive refinement, which describes the ideal situation where no information needs to be discarded for adapting an IB-optimal representation's granularity. Thanks in particular to a new geometric characterisation, we analytically derive the successive refinability of some specific IB problems (for binary variables, for jointly Gaussian variables, and for the relevancy variable being a deterministic function of the source variable), and provide a linear-programming-based tool to numerically investigate, in the discrete case, the successive refinement of the IB. We then soften this notion into a quantification of the loss of information optimality induced by several-stage processing through an existing measure of unique information. Simple numerical experiments suggest that this quantity is typically low, though not entirely negligible. These results could have important implications for (i) the structure and efficiency of incremental learning in biological and artificial agents, (ii) the comparison of IB-optimal observation channels in statistical decision problems, and (iii) the IB theory of deep neural networks.

Emergence of common concepts, symmetries and conformity in agent groups-an information-theoretic model.

The paper studies principles behind structured, especially symmetric, representations through enforced inter-agent conformity. For this, we consider agents in a simple environment who extract individual representations of this environment through an information maximization principle. The representations obtained by different agents differ in general to some extent from each other. This gives rise to ambiguities in how the environment is represented by the different agents. Using a variant of the information bottleneck principle, we extract a 'common conceptualization' of the world for this group of agents. It turns out that the common conceptualization appears to capture much higher regularities or symmetries of the environment than the individual representations. We further formalize the notion of identifying symmetries in the environment both with respect to 'extrinsic' (birds-eye) operations on the environment as well as with respect to 'intrinsic' operations, i.e. subjective operations corresponding to the reconfiguration of the agent's embodiment. Remarkably, using the latter formalism, one can re-wire an agent to conform to the highly symmetric common conceptualization to a much higher degree than an unrefined agent; and that, without having to re-optimize the agent from scratch. In other words, one can 're-educate' an agent to conform to the de-individualized 'concept' of the agent group with comparatively little effort.

Skilled motor control of an inverted pendulum implies low entropy of states but high entropy of actions.

The mastery of skills, such as balancing an inverted pendulum, implies a very accurate control of movements to achieve the task goals. Traditional accounts of skilled action control that focus on either routinization or perceptual control make opposite predictions about the ways we achieve mastery. The notion of routinization emphasizes the decrease of the variance of our actions, whereas the notion of perceptual control emphasizes the decrease of the variance of the states we visit, but not of the actions we execute. Here, we studied how participants managed control tasks of varying levels of difficulty, which consisted of controlling inverted pendulums of different lengths. We used information-theoretic measures to compare the predictions of alternative accounts that focus on routinization and perceptual control, respectively. Our results indicate that the successful performance of the control task strongly correlates with the decrease of state variability and the increase of action variability. As postulated by perceptual control theory, the mastery of skilled pendulum control consists in achieving stable control of goals by flexible means.

A space of goals: the cognitive geometry of informationally bounded agents.

Traditionally, Euclidean geometry is treated by scientists as a priori and objective. However, when we take the position of an agent, the problem of selecting a best route should also factor in the abilities of the agent, its embodiment and particularly its cognitive effort. In this paper, we consider geometry in terms of travel between states within a world by incorporating information processing costs with the appropriate spatial distances. This induces a geometry that increasingly differs from the original geometry of the given world as information costs become increasingly important. We visualize this 'cognitive geometry' by projecting it onto two- and three-dimensional spaces showing distinct distortions reflecting the emergence of epistemic and information-saving strategies as well as pivot states. The analogies between traditional cost-based geometries and those induced by additional informational costs invite a generalization of the notion of geodesics as cheapest routes towards the notion of infodesics. In this perspective, the concept of infodesics is inspired by the property of geodesics that, travelling from a given start location to a given goal location along a geodesic, not only the goal, but all points along the way are visited at optimal cost from the start.

Embodiment and Its Influence on Informational Costs of Decision Density-Atomic Actions vs. Scripted Sequences.

The importance of embodiment for effective robot performance has been postulated for a long time. Despite this, only relatively recently concrete quantitative models were put forward to characterize the advantages provided by a well-chosen embodiment. We here use one of these models, based on the concept of relevant information, to identify in a minimalistic scenario how and when embodiment affects the decision density. Concretely, we study how embodiment affects information costs when, instead of atomic actions, scripts are introduced, that is, predefined action sequences. Their inclusion can be treated as a straightforward extension of the basic action space. We will demonstrate the effect on informational decision cost of utilizing scripts vs. basic actions using a simple navigation task. Importantly, we will also employ a world with "mislabeled" actions, which we will call a "twisted" world. This is a model which had been used in an earlier study of the influence of embodiment on decision costs. It will turn out that twisted scenarios, as opposed to well-labeled ("embodied") ones, are significantly more costly in terms of relevant information. This cost is further worsened when the agent is forced to lower the decision density by employing scripts (once a script is triggered, no decisions are taken until the script has run to its end). This adds to our understanding why well-embodied (interpreted in our model as well-labeled) agents should be preferable, in a quantifiable, objective sense.

Space Emerges from What We Know-Spatial Categorisations Induced by Information Constraints.

Seeking goals carried out by agents with a level of competency requires an "understanding" of the structure of their world. While abstract formal descriptions of a world structure in terms of geometric axioms can be formulated in principle, it is not likely that this is the representation that is actually employed by biological organisms or that should be used by biologically plausible models. Instead, we operate by the assumption that biological organisms are constrained in their information processing capacities, which in the past has led to a number of insightful hypotheses and models for biologically plausible behaviour generation. Here we use this approach to study various types of spatial categorizations that emerge through such informational constraints imposed on embodied agents. We will see that geometrically-rich spatial representations emerge when agents employ a trade-off between the minimisation of the Shannon information used to describe locations within the environment and the reduction of the location error generated by the resulting approximate spatial description. In addition, agents do not always need to construct these representations from the ground up, but they can obtain them by refining less precise spatial descriptions constructed previously. Importantly, we find that these can be optimal at both steps of refinement, as guaranteed by the successive refinement principle from information theory. Finally, clusters induced by these spatial representations via the information bottleneck method are able to reflect the environment's topology without relying on an explicit geometric description of the environment's structure. Our findings suggest that the fundamental geometric notions possessed by natural agents do not need to be part of their a priori knowledge but could emerge as a byproduct of the pressure to process information parsimoniously.

Tilting Together: An Information-Theoretic Characterization of Behavioral Roles in Rhythmic Dyadic Interaction.

Every joint collaborative physical activity performed by a group of people, e.g., carrying a table, typically leads to the emergence of spatiotemporal coordination of individual motor behavior. Such interpersonal coordination can arise solely based on the observation of the partners' and/or the object's movements, without the presence of verbal communication. In this paper, we investigate how the social coupling between two individuals in a collaborative task translates into measured objective and subjective performance indicators recorded in two different studies. We analyse the trends in the dyadic interrelationship based on the information-theoretic measure of transfer entropy and identify emerging leader-follower roles. In our experimental paradigm, the actions of the pair of subjects are continuously and seamlessly fused, resulting in a joint control of an object simulated on a tablet computer. Subjects need to synchronize their movements with a 90° phase difference in order to keep the object (a ball) rotating precisely on a predefined circular or elliptic trajectory on a tablet device. Results demonstrate how the identification of causal dependencies in this social interaction task could reveal specific trends in human behavior and provide insights into the emergence of social sensorimotor contingencies.

Measuring Time with Minimal Clocks.

Being able to measure time, whether directly or indirectly, is a significant advantage for an organism. It allows for timely reaction to regular or predicted events, reducing the pressure for fast processing of sensory input. Thus, clocks are ubiquitous in biology. In the present article, we consider minimal abstract pure clocks in different configurations and investigate their characteristic dynamics. We are especially interested in optimally time-resolving clocks. Among these, we find fundamentally diametral clock characteristics, such as oscillatory behavior for purely local time measurement or decay-based clocks measuring time periods on a scale global to the problem. We include also sets of independent clocks (clock bags), sequential cascades of clocks, and composite clocks with controlled dependence. Clock cascades show a condensation effect, and the composite clock shows various regimes of markedly different dynamics.

Expanding the Active Inference Landscape: More Intrinsic Motivations in the Perception-Action Loop.

Active inference is an ambitious theory that treats perception, inference, and action selection of autonomous agents under the heading of a single principle. It suggests biologically plausible explanations for many cognitive phenomena, including consciousness. In active inference, action selection is driven by an objective function that evaluates possible future actions with respect to current, inferred beliefs about the world. Active inference at its core is independent from extrinsic rewards, resulting in a high level of robustness across e.g., different environments or agent morphologies. In the literature, paradigms that share this independence have been summarized under the notion of intrinsic motivations. In general and in contrast to active inference, these models of motivation come without a commitment to particular inference and action selection mechanisms. In this article, we study if the inference and action selection machinery of active inference can also be used by alternatives to the originally included intrinsic motivation. The perception-action loop explicitly relates inference and action selection to the environment and agent memory, and is consequently used as foundation for our analysis. We reconstruct the active inference approach, locate the original formulation within, and show how alternative intrinsic motivations can be used while keeping many of the original features intact. Furthermore, we illustrate the connection to universal reinforcement learning by means of our formalism. Active inference research may profit from comparisons of the dynamics induced by alternative intrinsic motivations. Research on intrinsic motivations may profit from an additional way to implement intrinsically motivated agents that also share the biological plausibility of active inference.

Cooperation and antagonism in information exchange in a growth scenario with two species.

We consider a simple information-theoretic model of communication, in which two species of bacteria have the option of exchanging information about their environment, thereby improving their chances of survival. For this purpose, we model a system consisting of two species whose dynamics in the world are modelled by a bet-hedging strategy. It is well known that such models lend themselves to elegant information-theoretical interpretations by relating their respective long-term growth rate to the information the individual species has about its environment. We are specifically interested in modelling how this dynamics are affected when the species interact cooperatively or in an antagonistic way in a scenario with limited resources. For this purpose, we consider the exchange of environmental information between the two species in the framework of a game. Our results show that a transition from a cooperative to an antagonistic behaviour in a species results as a response to a change in the availability of resources. Species cooperate in abundance of resources, while they behave antagonistically in scarcity.

An Informational Study of the Evolution of Codes and of Emerging Concepts in Populations of Agents.

We consider the problem of the evolution of a code within a structured population of agents. The agents try to maximize their information about their environment by acquiring information from the outputs of other agents in the population. A naive use of information-theoretic methods would assume that every agent knows how to interpret the information offered by other agents. However, this assumes that it knows which other agents it observes, and thus which code they use. In our model, however, we wish to preclude that: It is not clear which other agents an agent is observing, and the resulting usable information is therefore influenced by the universality of the code used and by which agents an agent is listening to. We further investigate whether an agent that does not directly perceive the environment can distinguish states by observing other agents' outputs. For this purpose, we consider a population of different types of agents talking about different concepts, and try to extract new ones by considering their outputs only.

Zipf's Law: Balancing Signal Usage Cost and Communication Efficiency.

We propose a model that explains the reliable emergence of power laws (e.g., Zipf's law) during the development of different human languages. The model incorporates the principle of least effort in communications, minimizing a combination of the information-theoretic communication inefficiency and direct signal cost. We prove a general relationship, for all optimal languages, between the signal cost distribution and the resulting distribution of signals. Zipf's law then emerges for logarithmic signal cost distributions, which is the cost distribution expected for words constructed from letters or phonemes.

Bivariate measure of redundant information.

We define a measure of redundant information based on projections in the space of probability distributions. Redundant information between random variables is information that is shared between those variables. But, in contrast to mutual information, redundant information denotes information that is shared about the outcome of a third variable. Formalizing this concept, and being able to measure it, is required for the non-negative decomposition of mutual information into redundant and synergistic information. Previous attempts to formalize redundant or synergistic information struggle to capture some desired properties. We introduce a new formalism for redundant information and prove that it satisfies all the properties necessary outlined in earlier work, as well as an additional criterion that we propose to be necessary to capture redundancy. We also demonstrate the behavior of this new measure for several examples, compare it to previous measures, and apply it to the decomposition of transfer entropy.

Perception-action loops of multiple agents: informational aspects and the impact of coordination.

Embodied agents can be conceived as entities perceiving and acting upon an external environment. Probabilistic models of this perception-action loop have paved the way to the investigation of information-theoretic aspects of embodied cognition. This formalism allows (i) to identify information flows and their limits under various scenarios and constraints, and (ii) to use informational quantities in order to induce the self-organization of the agent's behavior without any externally specified drives. This article extends the perception-action loop formalism to multiple agents. The multiple-access channel model is presented and used to identify the relationships between informational quantities of two agents interacting in the same environment. The central question investigated in this article is the impact of coordination. Information-theoretic limits on what can be achieved with and without coordination are identified. For this purpose, different abstract channels are studied, along with a concrete example of agents interacting in space. It is shown that, under some conditions, self-organizing systems based on information-theoretic quantities have a tendency to spontaneously generate coordinated behavior. Moreover, in the perspective of engineering such systems to achieve specific tasks, these information-theoretic limits put constraints on the amount of coordination that is required to perform the task, and consequently on the mechanisms that underlie self-organization in the system.

Generation of tactile maps for artificial skin.

Prior research has shown that representations of retinal surfaces can be learned from the intrinsic structure of visual sensory data in neural simulations, in robots, as well as by animals. Furthermore, representations of cochlear (frequency) surfaces can be learned from auditory data in neural simulations. Advances in hardware technology have allowed the development of artificial skin for robots, realising a new sensory modality which differs in important respects from vision and audition in its sensorimotor characteristics. This provides an opportunity to further investigate ordered sensory map formation using computational tools. We show that it is possible to learn representations of non-trivial tactile surfaces, which require topologically and geometrically involved three-dimensional embeddings. Our method automatically constructs a somatotopic map corresponding to the configuration of tactile sensors on a rigid body, using only intrinsic properties of the tactile data. The additional complexities involved in processing the tactile modality require the development of a novel multi-dimensional scaling algorithm. This algorithm, ANISOMAP, extends previous methods and outperforms them, producing high-quality reconstructions of tactile surfaces in both simulation and hardware tests. In addition, the reconstruction turns out to be robust to unanticipated hardware failure.

Information: currency of life?

In biology, the exception is mostly the rule, and the rule is mostly the exception. However, recent results indicate that known universal concepts in biology such as the genetic code or the utilization of ATP as a source of energy may be complemented by a large class of principles based on Shannon's concept of information. The present position paper discusses various promising pathways toward the formulation of such generic informational principles and their relevance for the realm of biology.

Stigmergic gene transfer and emergence of universal coding.

We consider a simple information-theoretic model for evolutionary dynamics approaching the "coding threshold," where the capacity to symbolically represent nucleic acid sequences emerges in response to a change in environmental conditions. We study the conditions when a coupling between the dynamics of a "proto-cell" and its proto-symbolic representation becomes beneficial in terms of preserving the proto-cell's information in a noisy environment. In particular, we are interested in understanding the behavior at the "error threshold" level, which, in our case, turns out to be a whole "error interval." The useful coupling is accompanied by self-organization of internal processing, i.e., an increase in complexity within the evolving system. Second, we study whether and how different proto-cells can stigmergically share such information via a joint encoding, even if they have slightly different individual dynamics. Implications for the emergence of biological genetic code are discussed.

Keep your options open: an information-based driving principle for sensorimotor systems.

The central resource processed by the sensorimotor system of an organism is information. We propose an information-based quantity that allows one to characterize the efficiency of the perception-action loop of an abstract organism model. It measures the potential of the organism to imprint information on the environment via its actuators in a way that can be recaptured by its sensors, essentially quantifying the options available and visible to the organism. Various scenarios suggest that such a quantity could identify the preferred direction of evolution or adaptation of the sensorimotor loop of organisms.