495 research outputs found
Generating functional analysis of Minority Games with real market histories
It is shown how the generating functional method of De Dominicis can be used
to solve the dynamics of the original version of the minority game (MG), in
which agents observe real as opposed to fake market histories. Here one again
finds exact closed equations for correlation and response functions, but now
these are defined in terms of two connected effective non-Markovian stochastic
processes: a single effective agent equation similar to that of the `fake'
history models, and a second effective equation for the overall market bid
itself (the latter is absent in `fake' history models). The result is an exact
theory, from which one can calculate from first principles both the persistent
observables in the MG and the distribution of history frequencies.Comment: 39 pages, 5 postscript figures, iop styl
Adaptive Trade-offs in the use of Social and Personal Information
In this chapter we review the redictions arising from theoretical models and outline the current empirical support for several social learning strategies, focusing largely on our own experimental studies and other recent work (Laland 2004; Kendal et al. 2005; Galef 2006). We draw attention to adaptive trade-offs in the use of social and personal information. Laland (2004) distinguished between two classes of social learning strategy, “when” strategies, which dictate the circumstances under which individuals copy others, and “who” strategies which specify from whom individuals learn. We address each in turn
The Cavity Approach to Parallel Dynamics of Ising Spins on a Graph
We use the cavity method to study parallel dynamics of disordered Ising
models on a graph. In particular, we derive a set of recursive equations in
single site probabilities of paths propagating along the edges of the graph.
These equations are analogous to the cavity equations for equilibrium models
and are exact on a tree. On graphs with exclusively directed edges we find an
exact expression for the stationary distribution of the spins. We present the
phase diagrams for an Ising model on an asymmetric Bethe lattice and for a
neural network with Hebbian interactions on an asymmetric scale-free graph. For
graphs with a nonzero fraction of symmetric edges the equations can be solved
for a finite number of time steps. Theoretical predictions are confirmed by
simulation results. Using a heuristic method, the cavity equations are extended
to a set of equations that determine the marginals of the stationary
distribution of Ising models on graphs with a nonzero fraction of symmetric
edges. The results of this method are discussed and compared with simulations
On-Line Learning with Restricted Training Sets: An Exactly Solvable Case
We solve the dynamics of on-line Hebbian learning in large perceptrons
exactly, for the regime where the size of the training set scales linearly with
the number of inputs. We consider both noiseless and noisy teachers. Our
calculation cannot be extended to non-Hebbian rules, but the solution provides
a convenient and welcome benchmark with which to test more general and advanced
theories for solving the dynamics of learning with restricted training sets.Comment: 19 pages, eps figures included, uses epsfig macr
The Relativistic Hopfield network: rigorous results
The relativistic Hopfield model constitutes a generalization of the standard
Hopfield model that is derived by the formal analogy between the
statistical-mechanic framework embedding neural networks and the Lagrangian
mechanics describing a fictitious single-particle motion in the space of the
tuneable parameters of the network itself. In this analogy the cost-function of
the Hopfield model plays as the standard kinetic-energy term and its related
Mattis overlap (naturally bounded by one) plays as the velocity. The
Hamiltonian of the relativisitc model, once Taylor-expanded, results in a
P-spin series with alternate signs: the attractive contributions enhance the
information-storage capabilities of the network, while the repulsive
contributions allow for an easier unlearning of spurious states, conferring
overall more robustness to the system as a whole. Here we do not deepen the
information processing skills of this generalized Hopfield network, rather we
focus on its statistical mechanical foundation. In particular, relying on
Guerra's interpolation techniques, we prove the existence of the infinite
volume limit for the model free-energy and we give its explicit expression in
terms of the Mattis overlaps. By extremizing the free energy over the latter we
get the generalized self-consistent equations for these overlaps, as well as a
picture of criticality that is further corroborated by a fluctuation analysis.
These findings are in full agreement with the available previous results.Comment: 11 pages, 1 figur
Slowly evolving geometry in recurrent neural networks I: extreme dilution regime
We study extremely diluted spin models of neural networks in which the
connectivity evolves in time, although adiabatically slowly compared to the
neurons, according to stochastic equations which on average aim to reduce
frustration. The (fast) neurons and (slow) connectivity variables equilibrate
separately, but at different temperatures. Our model is exactly solvable in
equilibrium. We obtain phase diagrams upon making the condensed ansatz (i.e.
recall of one pattern). These show that, as the connectivity temperature is
lowered, the volume of the retrieval phase diverges and the fraction of
mis-aligned spins is reduced. Still one always retains a region in the
retrieval phase where recall states other than the one corresponding to the
`condensed' pattern are locally stable, so the associative memory character of
our model is preserved.Comment: 18 pages, 6 figure
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