Kekuatan Komputasi Jaringan Saraf Tiruan?


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Katakanlah kita memiliki jaringan saraf maju umpan tunggal dengan input k dan satu output. Ini menghitung fungsi dari , cukup mudah untuk melihat bahwa ini memiliki setidaknya kekuatan komputasi yang sama dengan A C 0 . Hanya untuk bersenang-senang, kita akan memanggil sekumpulan fungsi yang dapat dihitung oleh jaringan saraf lapis tunggal " N e u r a l ".{0,1}n{0,1}AC0Neural

AC0

AC0NeuralNekamurSebuahl=SEBUAHC0? Juga apakah kelas kompleksitas semacam ini telah dipelajari sebelumnya?


1
A note on terminology -- important information is how many hidden layers there are. Zero hidden layer neural network with one output is just a linear threshold function, and is often (confusingly) called one layer or two layer neural network/perceptron, depending on whether inputs/outputs are considered layers. Also, in AI literature, neural networks are typically defined in terms of sigmoid functions which means that input/outputs are real valued. One hidden layer networks are known to be universal approximators in a sense that any continuous function can be approximated arbitrarily close
Yaroslav Bulatov

Jawaban:


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There are some references I could find: General-purpose computation with neural networks: A survey of complexity theoretic results, 2003 and Counting hierarchies: polynomial time and constant depth circuits, 1993.

It appears that neural networks are considered threshold circuits; i.e. those circuits using MAJORITY gates. In (2) it is the case that a depth d neural network has complexity TCd0 (here's a link to link to complexity zoo entry about TC0).

Since TC0 containts ACC0, which is AC0 with arbitrary MOD gates, then AC0TC0. Also, it is mentioned in the zoo that such circuits with depth 3 are strictly more powerful than those of depth 2.

In On the computational power of sigmoid versus Boolean threshold circuits,1991 it is mentioned that for a constant depth d, Boolean and real-valued threshold circuits (with polynomially bounded weights) are essentially the same.


Interesting, thanks. This is what I was looking for. More interesting still is that TC0 has a complete problems ... hmm ...
gabgoh
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