TY - JOUR

T1 - Recurrent Neural Networks with Small Weights Implement Definite Memory Machines

AU - Hammer, B

AU - Tino, Peter

PY - 2003/8/1

Y1 - 2003/8/1

N2 - Recent experimental studies indicate that recurrent neural networks initialized with "small" weights are inherently biased toward definite memory machines (Tino, Cernansky, & Benuskova, 2002a, 2002b). This article establishes a theoretical counterpart: transition function of recurrent network with small weights and squashing activation function is a contraction. We prove that recurrent networks with contractive transition function can be approximated arbitrarily well on input sequences of unbounded length by a definite memory machine. Conversely, every definite memory machine can be simulated by a recurrent network with contractive transition function. Hence, initialization with small weights induces an architectural bias into learning with recurrent neural networks. This bias might have benefits from the point of view of statistical learning theory: it emphasizes one possible region of the weight space where generalization ability can be formally proved. It is well known that standard recurrent neural networks are not distribution independent learnable in the probably approximately correct (PAC) sense if arbitrary precision and inputs are considered. We prove that recurrent networks with contractive transition function with a fixed contraction parameter fulfill the so-called distribution independent uniform convergence of empirical distances property and hence, unlike general recurrent networks, are distribution independent PAC learnable.

AB - Recent experimental studies indicate that recurrent neural networks initialized with "small" weights are inherently biased toward definite memory machines (Tino, Cernansky, & Benuskova, 2002a, 2002b). This article establishes a theoretical counterpart: transition function of recurrent network with small weights and squashing activation function is a contraction. We prove that recurrent networks with contractive transition function can be approximated arbitrarily well on input sequences of unbounded length by a definite memory machine. Conversely, every definite memory machine can be simulated by a recurrent network with contractive transition function. Hence, initialization with small weights induces an architectural bias into learning with recurrent neural networks. This bias might have benefits from the point of view of statistical learning theory: it emphasizes one possible region of the weight space where generalization ability can be formally proved. It is well known that standard recurrent neural networks are not distribution independent learnable in the probably approximately correct (PAC) sense if arbitrary precision and inputs are considered. We prove that recurrent networks with contractive transition function with a fixed contraction parameter fulfill the so-called distribution independent uniform convergence of empirical distances property and hence, unlike general recurrent networks, are distribution independent PAC learnable.

UR - http://www.scopus.com/inward/record.url?scp=0042326343&partnerID=8YFLogxK

U2 - 10.1162/08997660360675080

DO - 10.1162/08997660360675080

M3 - Article

SN - 1530-888X

SN - 1530-888X

SN - 1530-888X

SN - 1530-888X

SN - 1530-888X

SN - 1530-888X

VL - 15

SP - 1897

EP - 1926

JO - Neural Computation

JF - Neural Computation

IS - 8

ER -