Information Theory, Inference, and Learning ... - Inference Group

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Copyright Cambridge University Press 2003. On-screen viewing permitted. Printing not permitted. http://www.cambridge.org/0521642981You can buy this book for 30 pounds or $50. See http://www.inference.phy.cam.ac.uk/mackay/itila/ for links.Problems to look at before Chapter 40⊲ Exercise 40.1. [2 ] What is ∑ N( NK=0 K)?[The symbol ( NK)means the combinationN!K!(N−K)! .]⊲ Exercise 40.2. [2 ] If the top row of Pascal’s triangle (which contains the singlenumber ‘1’) is denoted row zero, what is the sum of all the numbers inthe triangle above row N?⊲ Exercise 40.3. [2 ] 3 points are selected at random on the surface of a sphere.What is the probability that all of them lie on a single hemisphere?This chapter’s material is originally due to Polya (1954) and Cover (1965) andthe exposition that follows is Yaser Abu-Mostafa’s.482
Copyright Cambridge University Press 2003. On-screen viewing permitted. Printing not permitted. http://www.cambridge.org/0521642981You can buy this book for 30 pounds or $50. See http://www.inference.phy.cam.ac.uk/mackay/itila/ for links.40Capacity of a Single Neuron{t n } N n=1✲ Learningalgorithm ✲ w ✲ w✻✻✲ {ˆt n } N n=1Figure 40.1. Neural networklearning viewed ascommunication.{x n } N n=1{x n } N n=140.1 Neural network learning as communicationMany neural network models involve the adaptation of a set of weights w inresponse to a set of data points, for example a set of N target values D N ={t n } N n=1 at given locations {x n} N n=1 . The adapted weights are then used toprocess subsequent input data. This process can be viewed as a communicationprocess, in which the sender examines the data D N and creates a message wthat depends on those data. The receiver then uses w; for example, thereceiver might use the weights to try to reconstruct what the data D N was.[In neural network parlance, this is using the neuron for ‘memory’ rather thanfor ‘generalization’; ‘generalizing’ means extrapolating from the observed datato the value of t N+1 at some new location x N+1 .] Just as a disk drive is acommunication channel, the adapted network weights w therefore play therole of a communication channel, conveying information about the trainingdata to a future user of that neural net. The question we now address is,‘what is the capacity of this channel?’ – that is, ‘how much information canbe stored by training a neural network?’If we had a learning algorithm that either produces a network whose responseto all inputs is +1 or a network whose response to all inputs is 0,depending on the training data, then the weights allow us to distinguish betweenjust two sorts of data set. The maximum information such a learningalgorithm could convey about the data is therefore 1 bit, this information contentbeing achieved if the two sorts of data set are equiprobable. How muchmore information can be conveyed if we make full use of a neural network’sability to represent other functions?40.2 The capacity of a single neuronWe will look at the simplest case, that of a single binary threshold neuron. Wewill find that the capacity of such a neuron is two bits per weight. A neuronwith K inputs can store 2K bits of information.To obtain this interesting result we lay down some rules to exclude lessinteresting answers, such as: ‘the capacity of a neuron is infinite, because each483
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