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Assessing the Computational Power and Mechanisms of Information Processing of Reservoirs In many cases, artificial neural networks are created with a specific goal in mind, for example to approximate a particular function or system. Training success and computational capability of the network with respect to this task ... |
The loss on a specific class of problems does not express the general computational power of the network, though. This property becomes more interesting when a part of the system is used for more than one task: relevant cases would be dynamical reservoirs that are used for multiple applications, networks that are train... |
2.1 Information-theory related measures Information theory and (Shannon) entropy have been used in a number of ways in neural network and complex systems research. One particular heuristic to measure (and eventually improve) RNN is to estimate and influence the entropy distribution of firing rates of a neuron. In indiv... |
Due to the limited degrees of freedom of the approach, the desired target distribution cannot be approximated for every kind of input Boedecker et al., ( 2009 . Intrinsic plasticity as well as its particular limitation can be related to Ashby’s law of requisite variety Ashby, ( 1956 in that by increasing variety availa... |
The field of information dynamics Lizier et al., ( 2007 2012 provides information-theoretic measures that explicitly deal with processes or time-series. Information storage , as one of the tools, quantifies how much of the stored information is actually in use at the next time step when the next process value is comput... |
[EQUATION] Information transfer , expressed as transfer entropy Schreiber, ( 2000 , quantifies the influence of another process on the next state (for a formal definition, see Sect. 3.3 below). Boedecker et al., ( 2011 use these measures, to gain a better understanding of computational mechanisms inside the reservoir, ... |
Fisher information also plays a role in quantifying the memory stored in a dynamical reservoir: Information about the recent input in reservoir networks is stored in the transient dynamics, rather than in attractor states. To quantify this information storage, Ganguli et al., ( 2008 use Fisher information as basis for ... |
[EQUATION] The Fisher Memory Matrix (FMM) between the present state of the system [MATH] and the past signal is defined as [EQUATION] |
Diagonal elements [MATH] are the Fisher information that the system keeps in [MATH] about a pulse at [MATH] steps back in the past, i.e., the decay of the memory trace of a past input. [MATH] is called the Fisher memory curve (FMC). |
Tino and Rodan, ( 2013 investigate the relation between [MATH] and the short term memory capacity MC Jaeger, ( 2001 (details on MC in the following subsection), and show that the two are closely related in linear systems. For these, [MATH] is independent of the actual input used. In the general, nonlinear case that is ... |
A measure for Active Information Storage in input-driven systems has been proposed to quantify storage capabilities of a nonlinear system independent of particular inputs Obst et al., ( 2013 . The measure is an Active Information Storage Lizier et al., ( 2012 where the current input [MATH] is conditioned out: |
[EQUATION] The idea for this measure is to remove influences of structure in input data, and to only characterize the system itself, rather than a combination of system and input data. In theory, this influence would be removed by having the history sizes in computing the information storage converge to infinity. Large... |
2.2 Measures related to learning theory Legenstein and Maass, ( 2007 propose two measures to quantify the computational capabilities of reservoirs in the context of liquid state machines, one of the two main flavors of reservoir computing networks: the linear separation property und the generalization capability. The l... |
For the generalization ability, they propose to approximate the VC-dimension of class [MATH] of the reservoir, which includes all maps from a set [MATH] of inputs [MATH] into [MATH] which can be implemented by a reservoir [MATH] . They present a theorem (and corresponding proof sketch) stating that under the assumption... |
According to Legenstein and Maass, ( 2007 , a simple difference of both (normalized) measures leads to good predictions about which reservoirs perform well on a range of tasks. |
The loss or the success on a set of test functions is another possibility to characterize the systems from a learning point of view. One such measure is the short term memory capacity MC Jaeger, ( 2001 that we briefly mentioned above. To compute the MC, a network is trained to generate delayed versions [MATH] of a sing... |
[EQUATION] The symbols [MATH] and [MATH] denote covariance and variance, respectively. Each coefficient takes values between 0 and 1, and expresses how much of the variance in one signal is explainable by the other. As shown in Jaeger, ( 2001 , for i.i.d. input and linear output units, the MC of [MATH] -unit RNN is bou... |
Another approach in this area is the information processing capacity of a dynamical system Dambre et al., ( 2012 . It is a measure based on the mean square error MSE in reconstructing a set of functions [MATH] . The idea is to distinguish from approaches that view dynamical systems merely providing some form of memory ... |
[EQUATION] This computed capacity is dependent on the input. In order to avoid an influence of structure in the input on the results, i.i.d. input is required for the purpose of measuring the capacity. To measure information processing capacity, several functions [MATH] have to be evaluated. The idea is that if [MATH] ... |
The proposed approach has been used to compare different implementations of dynamical systems, like reaction-diffusion systems and reservoirs. An interesting idea that is also mentioned in Dambre et al., ( 2012 would be to extend the approach so that the underlying system adapts to provide specific mappings. One possib... |
2.3 Measures related to dynamical systems theory To gain understanding of the internal operations that enable high-dimensional RNNs solving a given task, a recent effort by Sussillo and Barak, ( 2013 draws on tools from dynamical systems theory. Using numerical optimization techniques, the authors identify fixed points... |
In Williams and Beer, ( 2010 , the authors argue for a complementary role of dynamical analysis, which involves, e.g., looking at attractors and switching between attractor states, and also an information-theoretic analysis when trying to understand computation in dynamical systems (including input-driven ones – even t... |
that a state variable carries about a particular stimulus at each time step. The behavior of the agent can then be explained by a sudden gain and then loss of information about object sizes in the first neuron, and then a rapid gain of information about relative size of the objects. In summary, the authors state that t... |
Another approach from dynamical systems theory to understand and predict computational capabilities in RNNs builds on the concept of Lyapunov exponents. Although these concepts are only defined for autonomous dynamical systems, an analogous idea is to introduce a small perturbation into the state of one of two identica... |
Improving Reservoir Information Processing Capabilities Through Self-Organized Adaptation A pragmatic way to evaluate the quality of a reservoir is to train the output, and evaluate it on a training or validation set Lukoševičius, 2012a . In most circumstances, training is fast so that a number of hyper-parameter setti... |
3.1 SORN: self-organized optimization based on 3 local plasticity mechanisms One approach that has demonstrated how self-organization can be leveraged to optimize a reservoir network can be found in Lazar et al., ( 2009 . SORN is a self-organizing recurrent network architecture using discrete-time binary units. The thr... |
[EQUATION] [MATH] and [MATH] are threshold values, drawn randomly from positive intervals for excitatory units and inhibitory units, respectively. [MATH] is the heaviside step function, and [MATH] the network input drive. Matrices [MATH] and [MATH] are fully connected, and represent connections between inhibitory and e... |
STDP and synaptic scaling update connections of excitatory units of the reservoir, while IP changes their thresholds. Inhibitory neurons and their connections remain unchanged. In the SORN the STDP for some small learning constant [MATH] is formalized as: |
[EQUATION] Synaptic scaling normalizes the values to sum up to one: [EQUATION] IP learning is responsible for spreading activations more evenly, using a learning rate [MATH] , and a target firing rate of [MATH] |
[EQUATION] Lazar et al., ( 2009 show that the SORN outperforms static reservoir networks using a letter prediction task. The network has to predict the next letter in a sequence of two different artificial words of length [MATH] . These words are made up of three different characters, with the second character repeated... |
The combination of the three mechanisms appears to be a key to successful self-organization in an RNN. Figure illustrates that the dynamics of SORN reservoir become sub-optimal if only two of the three plasticity mechanisms are active. Without synaptic normalization, the network units become highly synchronized. This s... |
Though some of the self-organizing mechanisms like STDP are biologically plausible, there are not too many examples of successful applications for training RNNs, or, as Lazar et al., ( 2009 states, “Understanding and controlling the ensuing self-organization of network structure and dynamics as a function of the networ... |
3.2 Hierarchical Self-Organizing Reservoirs A different approach based on self-organized optimization of reservoirs is presented in Lukoševičius, 2012b . The author compares classical ESNs and recurrent RBF-unit based reservoir networks (called Self-Organizing Reservoirs, SORs) which resemble Recurrent Self-Organizing ... |
The update equations for the SOR are: [EQUATION] Here, the internal reservoir neuron states at time [MATH] are collected in vector [MATH] and their update in vector [MATH] . The factor [MATH] is the leak-rate. The vector [MATH] contains the input-signal, while matrices [MATH] and [MATH] are the input and recurrent weig... |
The unsupervised training of the SOR updates the input and recurrent weights as: [EQUATION] where [MATH] and [MATH] is a time-dependent learning rate. The learning-gradient distribution function [MATH] is defined either as: |
[EQUATION] where [MATH] is the distance between reservoir units [MATH] and [MATH] on a specific topology, [MATH] is a function returning the index of a best matching unit (BMU), and [MATH] is the time-dependent of the learning gradient distribution. With this definition of [MATH] , the learning proceeds according to th... |
[EQUATION] where [MATH] denotes the index of node [MATH] in the descending ordering of activities [MATH] (see Lukoševičius, 2012b for additional details). Both algorithms were found to be similarly effective to improve the pattern separation capability of reservoirs compared to standard ESNs when tested on detection of... |
3.3 Guided Self-Organization of Reservoir Information Transfer In Obst et al., ( 2010 , the information transfer between input data and desired outputs is used to guide the adaptation of the self-recurrence in the hidden layer of a reservoir computing network. The idea behind this step is to change the memory within th... |
The network dynamics is updated as: [EQUATION] where [MATH] are the unit activations, [MATH] is the [MATH] reservoir weight matrix, [MATH] the input weight vector, [MATH] a vector of local decay factors, [MATH] is the identity matrix, and [MATH] denotes the discrete time step. As a nonlinearity, [MATH] is used. The [MA... |
[EQUATION] where [MATH] represents the memory length of unit [MATH] [MATH] ), initialized to [MATH] . Increasing individual [MATH] through adaptation increases the influence of a unit’s past states on its current state. The information transfer is quantified as a conditional mutual information or transfer entropy Schre... |
[EQUATION] Parameters [MATH] and [MATH] are history sizes, which lead to finite-sized approximations of the transfer entropy for finite values. |
In a first step, the required history size [MATH] is determined which maximizes the information transfer [MATH] from input [MATH] to output [MATH] . This value will increase for successively larger history sizes, but the increases are likely to level off for large values of [MATH] . Therefore, [MATH] is determined as t... |
[EQUATION] In a second step, the local couplings of the reservoir units are adapted so that the transfer entropy from the input of each unit to its respective output is optimized for the particular input history length [MATH] , as determined in step one. Over each epoch [MATH] of length [MATH] , we compute the transfer... |
[EQUATION] If the information transfer during the current epoch [MATH] exceeds the information transfer during the past epoch by a threshold (i.e., [MATH] ), the local memory length [MATH] is increased by one. In case [MATH] , the local memory length is decreased by one, down to a minimum of [MATH] . The decay factors ... |
In Obst et al., ( 2010 , the method is tested on a one-step ahead prediction of unidirectionally coupled maps and of the Mackey-Glass time series benchmark. Showing results for the former task as an example, Figure (left) displays the mean square errors of the prediction over the test data for different coupling streng... |
Quantifying task complexity Most currently existing measures capture some of the generic computational properties of recurrent neural networks (as an important class of input-driven system), such as memory capacity or entropy at the neuron-level, but do not take task complexity into account. Optimization of the network... |
It is possible to use some of the tools that we introduced above, and take an information-theoretic approach to tackle this problem. Essentially we are interested in quantifying how difficult it is for a system to produce its next output. The systems we are interested in take a time series [MATH] as an input, and have ... |
The Active Information Storage [MATH] Lizier et al., ( 2012 can be used to capture the influence of previous outputs in producing the next output: how much information is contained in the past of [MATH] that can be used to compute its next state? This is expressed as the average mutual information between past [MATH] o... |
[EQUATION] We use [MATH] to represent finite- [MATH] estimates. Now, [MATH] and [MATH] allow for two kind of measurements: (a) higher values for [MATH] indicate better predictability of [MATH] from its own past, i.e., [MATH] is one component of the overall task difficulty. (b) With increasing values of [MATH] , estimat... |
The other component that plays a role in the task is the input [MATH] . Its contribution to producing the next output [MATH] of the system, too, can be quantified, using the transfer entropy introduced above. The transfer entropy indicates how much information the input [MATH] contributes to the next state [MATH] , giv... |
Unfortunately, using these quantities to compare tasks or to design systems is not entirely straightforward, for a number of reasons. For continuous-valued time series [MATH] and [MATH] , estimating mutual informations is cumbersome, and requires larger amounts of data in particular for larger history sizes [MATH] and ... |
We will reserve a detailed investigation of applying both measures to a later publication. As a concept to explain contributions of input and output history, they can be an indicator for how complex the information processing system needs to be. It will also be interesting to see how other measures relate to them, and ... |
As another example, measuring the individual distributions of unit activations in the reservoir and their divergence from a maximum entropy distribution capture properties of the input combined with properties of the network. On the other hand, Active Information Storage for input-driven systems, applied to reservoirs ... |
Related work on complexity measures includes Grassberger’s forecast complexity Grassberger, ( 1986 2012 , which considers the difficulty of making an optimal prediction of a sequence created by a stochastic process. A sequence can be compressed up to its entropy rate, and the forecast complexity is the computational co... |
Conclusion We presented methods to assess different computational properties of input-driven RNNs, and reservoir computing networks in particular, in the first part of the paper. These methods were drawn from information-theory, statistical learning theory, and dynamical systems theory, and provided different perspecti... |
In the second part of the paper, we presented some recent efforts at implementing self-organized optimization for reservoir computing networks. One approach combined different plasticity mechanisms to improve coding quality and separation ability of the network, while a different approach was using methods similar to r... |
As a next step towards methods that are able to automatically generate or optimize recurrent neural networks for a specific task (or class of tasks), it seems worthwhile to combine measures for network properties and task complexity, and devise algorithms that adjust the former based on the latter. The approach taken i... |
A comparison of how the measures of the information dynamics framework Lizier et al., ( 2007 2012 , the information processing capacity for dynamical systems Dambre et al., ( 2012 , measures of criticality Bertschinger and Natschläger, ( 2004 ); Prokopenko et al., ( 2011 or of memory capacity Jaeger, ( 2001 ); Ganguli ... |
# Source: arxiv 1309.2959 # Title: Self-Organization In 1-d Swarm Dynamics # Sections: all # Downloaded: 2026-03-03T01:58:49.029606+00:00 |
Self-Organization In 1-d Swarm Dynamics Abstract Self-organization of a biologically motivated swarm into smaller subgroups of different velocities is found by solving a 1-dimensional adaptive-velocity swarm, in which the velocity of an agent is averaged over a finite local radius of influence. Using a mean field model... |
Key words: self-organization of swarm, phase space, multi-agent system, dynamical system, group-division. PACS:89.75.Fb, 05.65.+b |
Introduction The dynamics of swarm behavior has long been a mystery in nature , and despite intensive work, remains an open problem today |
. Examples of swarms include flocks of birds, schools of fishes and even crowds of people in cramped public spaces. This research effort has introduced a wide range of models for swarm behavior which is the background for our model in this paper. In 1986, Reynolds |
simulated swarm behavior according to the following rules: move in the same direction as the neighbors; remain close to the neighbors; avoid collisions with neighbors. In 1995, Vicsek et al. |
proposed an important simpler model, in which all the agents have a constant speed and only change their headings according to the other agents in the influence radius (cf. also |
). Other models for swarm behavior include . In 1995, Toner and Tu proposed a non-equilibrium continuous dynamical model for the collective motion of large groups of biological organisms. It describes a class of microscopic rules, which includes the model in |
as a special case. In 2003, Aldana and Huepe investigated the conditions that produce a phase transition from an ordered to a disordered state in a model of two-dimensional agents with a ferromagnetic-like interaction. In their model, the agents still keep a constant speed and change their headings as in |
. However, they are only influenced by their neighbors in a fixed radius. Besides, Morale et al have proposed stochastic models for swarms from 2000. These models are based on a number of individuals subject to several distinct mechanisms simultaneously - long range attraction, and short range repulsion, in addition to... |
In 2008, Nabet, Benjamin, et al. established a model simpler than , in which the agent speed is fixed and the effects of agent position are ignored. The swarm in their model includes two informed subgroups which have preferred directions of motion and a third naive group that does not have a preference. They investigat... |
In 2011, W.Li and X.Wang proposed another model which is also based on . In their model, each agent adjusts its heading and speed simultaneously according to its local neighbors. The change in speed at each time step depends only on the degree of local direction consensus. Unlike our model, the new speed does not depen... |
Motivated by the aim of a more complete understanding of swarm behavior in the real world, we introduce a new model in 1-dimensional space, in which each agent continuously updates its speed and direction according to the average velocity of the agents within its influence radius. We simulate the model with initial pos... |
The paper is organized as follows. In section 2, we formulate the discrete and mean field models in this paper. In section 3, we discuss simulations conducted using these models to find the effects of typical scales of initial position and velocity on the group-division phenomenon. Then we examine the energetics of thi... |
models 2.1 Discrete model We consider a swarm of [MATH] agents [MATH] in 1-d space, each of which has velocity [MATH] and position [MATH] , and assume that every agent can sense the other agents within the fixed radius [MATH] . We call these agents its neighbors and assume each agent adjusts its velocity according to t... |
[EQUATION] We also have: [EQUATION] Together, we get the equations in 2 dimensional phase space given by position and velocity. Simulation of this model in two-dimensional phase space, as shown in Figure.1, reveals a group-division phenomenon in the dynamics. We measure the typical scales of initial position and veloci... |
[EQUATION] and [EQUATION] Then we generate the initial swarm using Gaussian distribution with typical scales [MATH] and [MATH] respectively. We run this model 20000 periods for 1000 agents, in each period, all the agents adjust their velocity in response to their neighbors. At the beginning, there is only one huge grou... |
2.2 Mean field model We use a mean field method to build a model in phase space. Assume the number of agents in the swarm is large enough; also the agents in any small area in phase space is well-distributed. Let [MATH] be the velocity and position as before. Define [MATH] as the probability measure in phase space — fo... |
[EQUATION] where [MATH] denotes a ball of radius [MATH] centered at point [MATH] Let [MATH] denote the time, [MATH] denote the influencing radius. Define [MATH] as the flow velocity in phase space, and define [MATH] . Then |
[EQUATION] where [MATH] denotes the expectation. Moreover: [EQUATION] Let [MATH] be a random variable which denote the velocity of a agent in [MATH] [MATH] , where [MATH] is the random variable which denote the number of neighbors. Since |
[EQUATION] thus [EQUATION] Therefore, we have the mean field equation for the physical velocity: [EQUATION] and the phase space velocity: |
[EQUATION] Finally, by conservation law in phase space: [EQUATION] Using the finite volume method under periodic boundary condition, we get the numerical solution of this integro-differential equation ( ). As shown in Figure.2, the numerical solution confirms the group-division phenomenon we find in the discrete model. |
Effects of typical scales of initial position and velocity on group-division phenomenon As shown above, the mean field model matches the discrete simulation very well. To see this more clearly, we display the data projection on the position axis. An example of such a projection, shown in Figure.3, reveals the group-div... |
Next, we find that the group-division phenomenon is related to the typical scales of initial position and velocity, as well as the influence radius. Scale [MATH] and [MATH] with [MATH] , we get some further examples, which are shown in the following (Figure.4). Y-axis indicates [MATH] , the ratio of the size of largest... |
Energy analysis In this section, we analyze energetics of the group-division phenomenon. We find the energy-time curve has two stages connected by a sharp transition(see figure.5). |
Define [MATH] as the total kinetic energy of the system. Under the group-division phenomenon, label each group as: [MATH] , and the corresponding velocities of the groups as: [MATH] [MATH] defines the number of agents in group [MATH] . Thus: |
[EQUATION] According to the last line ( ), we write the kinetic energy into three terms, where [MATH] is the first term, [MATH] is the second term, and [MATH] is the third term. |
The first term is the relative kinetic energy of the agents with respect to the center of mass of corresponding groups; the second term is the relative kinetic energy of the groups with respect to the global center of mass; the third term is the kinetic energy of the global center of mass, which is almost constant from... |
At the beginning of the simulation, since we only have one large group: [EQUATION] At the end of the simulation, since we have several groups, and each agent in its group will have the same velocity (i.e.: [MATH] ): |
[EQUATION] Therefore, the kinetic energy shifts from the first term to the second term, as it decreases monotonically from its initial value. |
Moreover, since: [EQUATION] solving this ODE yields: [EQUATION] Therefore the first term could be written as: [EQUATION] We consider the change of the variance of agent velocity, which can be interpreted as kinetic energy. By equation ( ), we know that the kinetic energy decrease exponentially in the first stage, which... |
will finally be constant, the curve of kinetic energy should be a straight line after division is complete, so the log of the variance is still a straight line after division. Since we know that the kinetic energy shifts from the first term to the second term, we pick the point of the largest change in the semi-log var... |
During the process, the variance of velocity decreases exponentially (see Figure.5). We take the log of the variance of [MATH] (Y-axis in left figure), and consider its rate of change of slope. The maximum point of change of slope rate, which we take to be the end of division, is 195 (258) of the mean field model (disc... |
Conclusion In our paper, we propose an adaptive-velocity swarm model in which each agent not only adjusts its movement direction but also adjusts its speed as a function of its local neighbors in 1-dimensional space. Such a spatially 1-dimensional model is nonetheless capable of supporting a robust group-division pheno... |
Some difficult but important problems of our model remain to be further investigated. For example, under what condition can we infer the same result in 2-dimensional and 3-dimensional space? What kind of behavior will emerge if certain agents have greater influence than others? In the more applied aspect, how do we con... |
Acknowledgements This work was supported in part by the Army Research Office Grants No. W911NF-09-1-0254 and W911NF-12-1-0546. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Resea... |
# Source: arxiv 1309.6261 # Title: Chaos and Unpredictability in Evolution # Sections: all # Downloaded: 2026-03-03T01:59:14.380642+00:00 |
Chaos and Unpredictability in Evolution Abstract The possibility of complicated dynamic behaviour driven by non-linear feedbacks in dynamical systems has revolutionized science in the latter part of the last century. Yet despite examples of complicated frequency dynamics, the possibility of long-term evolutionary chaos... |
Evolution, Adaptive dynamics, Chaos Author Summary 40 years ago, the discovery of deterministic chaos has revolutionized science. Surprisingly, few of these insights have entered the realm of evolutionary biology, where “survival of the fittest” epitomizes evolution as an optimization process that generally converges t... |
II Introduction Evolution generally takes place in complex ecosystems and is affected by many different processes that generate non-linear dependencies. According to general dynamical systems theory, which has shown that even simple dynamical system can exhibit complicated dynamics Li & Yorke ( 1975 ); May ( 1976 ); Ba... |
Static fitness landscapes describe frequency-independent selection whose strength and direction is not affected by the current phenotypic composition of an evolving population. However, it is widely recognized that ecological interactions, such as competition and predation, often lead to frequency-dependent selection, ... |
It is important to distinguish models for short-term frequency dynamics from evolutionary models in which the dynamic variables are the (mean) phenotypes themselves, and which track the trajectories of such phenotypes in continuous phenotype spaces and over long evolutionary time scales. The phase space for this type o... |
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