Analysis of Learning from Stochastic Rules in the Framework of Replica Symmetry Breaking

概要

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We study learning from examples by a perceptron when a return to an example is given by astochastic relation which is represented by the conditional probability distribution P('u), where uis proportional to the inner product between the optimal synaptic weight vector and the examplevector. The problem is analyzed by replica method in the case of spherical weights. Since thereplica symmetric(RS) solution turns out to be unstable for stochastic cases, we consider theone-step replica symmetry breaking (RSB) solution, We investigate the asymptotic behavior ofthe learning curve as a : -% -* oc, where p is the number of samples and N is the dimension ofthe synaptic weights. The average generalization error c. is expressed as (e. - (min) oc a ' in theasymptotic region. For the minimum-error algorithm, we find 7 : $ for the RS solution and7 : $ for the one-step RSB solution in the case of the function P('u) which is monotonic andexpressed as P(u) :const.-;a sgn(u)?'u?' around ct : 0. That is, the exponent 'y is determined bythe local property of P(u) around u : O. In particular, for the case of 6 : l, the one-step RSBansatz, unlike the RS ansatz, gives the same result up to a logarithmic correction as previousresults obtained by non-replica methods in the one-dimensional model.
社団法人日本物理学会の論文
1996-12-15