Influence of external information in the minority game

The influence of a fixed number of agents with the same fixed behavior on the dynamics of the minority game is studied. Alternatively, the system studied can be considered the minority game with a change in the comfort threshold away from half filling. Agents in the frustrated, non ergodic phase tend to overreact to the information provided by the fixed agents, leading not only to large fluctuations, but to deviations of the average occupancies from their optimal values. Agents which discount their impact on the market, or which use individual strategies reach equilibrium states, which, unlike in the absence of the external information provided by the fixed agents, do not give the highest payoff to the collective.Comment: 8 pages, 6 figure

Dynamical and Stationary Properties of On-line Learning from Finite Training Sets

The dynamical and stationary properties of on-line learning from finite training sets are analysed using the cavity method. For large input dimensions, we derive equations for the macroscopic parameters, namely, the student-teacher correlation, the student-student autocorrelation and the learning force uctuation. This enables us to provide analytical solutions to Adaline learning as a benchmark. Theoretical predictions of training errors in transient and stationary states are obtained by a Monte Carlo sampling procedure. Generalization and training errors are found to agree with simulations. The physical origin of the critical learning rate is presented. Comparison with batch learning is discussed throughout the paper.Comment: 30 pages, 4 figure

Continuum time limit and stationary states of the Minority Game

We discuss in detail the derivation of stochastic differential equations for the continuum time limit of the Minority Game. We show that all properties of the Minority Game can be understood by a careful theoretical analysis of such equations. In particular, i) we confirm that the stationary state properties are given by the ground state configurations of a disordered (soft) spin system; ii) we derive the full stationary state distribution; iii) we characterize the dependence on initial conditions in the symmetric phase and iv) we clarify the behavior of the system as a function of the learning rate. This leaves us with a complete and coherent picture of the collective behavior of the Minority Game. Strikingly we find that the temperature like parameter which is introduced in the choice behavior of individual agents turns out to play the role, at the collective level, of the inverse of a thermodynamic temperature.Comment: Revised version (several new results added). 12 pages, 5 figure

Supervised Learning with Restricted Training Sets: a Generating Functional Analysis

. We study the dynamics of supervised on-line learning of realizable tasks in feed-forward neural networks. We focus on the regime where the number of examples used for training is proportional to the number of input channels N . Using generating function techniques from spin glass theory, we are able to average over the composition of the training set and transform the problem for N ## to an e#ective single pattern system, described completely by the student autocovariance, the student-teacher overlap and the student response function, with exact closed equations. Our method applies to arbitrary learning rules, i.e. not necessarily of a gradient-descent type. The resulting exact macroscopic dynamical equations can be integrated without finite-size e#ects up to any degree of accuracy, but their main value is in providing an exact and simple starting point for analytical approximation schemes. Finally, we show how, in the region of absent anomalous response and using the hypothesis th..