What can a network represent

#Preliminary

Perceptron

  • Threshold unit
    • β€œFires” if the weighted sum of inputs exceeds a threshold
  • Soft perceptron
    • Using sigmoid function instead of a threshold at the output
    • Activation: The function that acts on the weighted combination of inputs (and threshold)
  • Affine combination
    • Different from Linear combination: the result of mapping zero is not zero.

Multi-layer perceptron

  • Depth
    • Is the length of the longest path from a source to a sink
    • Deep: Depth greater than 2
  • Inputs/Outputs are real or Boolean stimuli
  • What can this network compute?

#Universal Boolean functions

  • A perceptron can model any simple binary Boolean gate
    • Using weight 1 or -1 to model function
    • The universal AND gate: $(\bigwedge_{i=1}^{L} X_{i}) \wedge(\bigwedge_{i=L+1}^{N} \bar{X}_{i})$
    • The universal OR gate: $(\bigvee_{i=1}^{L} X_{i}) \vee(\bigvee_{i=L+1}^{N} \bar{X}_{i})$
    • Cannot compute an XOR
  • MLPs can compute the XOR
  • MLPs are universal Boolean functions

    • Can compute any Boolean function
  • A Boolean function is just a truth table

    • So expressed the result in disjunctive normal form, like

$$ \begin{aligned} Y=& \bar{X}_1 \bar{X}_2 X_3 X_4 \bar{X}_5+\bar{X}_1 X_2 \bar{X}_3 X_4 X_5+\bar{X}_1 X_2 X_3 \bar{X}_4 \bar{X}_5+X_1 \bar{X}_2 \bar{X}_3 \bar{X}_4 X_5+X_1 \bar{X}_2 X_3 X_4 X_5+X_1 X_2 \bar{X}_3 \bar{X}_4 X_5 \end{aligned} $$

  • In this case, need 5 neurons in the hidden layer.

#Need for depth

  • A one-hidden-layer MLP is a Universal Boolean Function

    • But the largest number of perceptrons is expontial: $2^N$
  • How about depth?

    • Will require $3(N-1)$ perceptrons, linear in $N$ to express the same function
    • Using associatable rules, can be arranged in $2\log_2 N$ layers
    • eg. model $O=W \oplus X \oplus Y \oplus Z$
  • The challenge of depth

    • Using only $K$ hidden layers will require $O(2^{CN})$ neurons in the $K$th layer, where $C = 2^{-(k-1)/2}$
    • A network with fewer than the minimum required number of neurons cannot model the function

#Universal classifiers

  • Composing complicated β€œdecision” boundaries
  • Using OR to create more decision boundaries
    • Can compose arbitrarily complex decision boundaries
    • Even using one-layer MLP

#Need for depth

  • A naïve one-hidden-layer neural network will required infinite hidden neurons
  • Construct basic unit and add more layers to decrese #neurons
  • The number of neurons required in a shallow network is potentially exponential in the dimensionality of the input

#Universal approximators

  • A one-layer MLP can model an arbitrary function of a single input
  • MLPs can actually compose arbitrary functions in any number of dimensions
    • Even without "activation"
  • Activation
    • A universal map from the entire domain of input values to the entire range of the output activation

#Optimal depth and width

  • Deeper networks will require far fewer neurons for the same approximation error
  • Sufficiency of architecture
    • Not all architectures can represent any function
  • Continuous activation functions result in graded output at the layer
    • To capture information "missed" by the lower layer

#Width vs. Activations vs. Depth

  • Narrow layers can still pass information to subsequent layers if the activation function is sufficiently graded
    • But will require greater depth, to permit later layers to capture patterns
  • Capacity of the network
    • Information or Storage: how many patterns can it remember
    • VC dimension: bounded by the square of the number of weights in the network
    • Straight forward: largest number of disconnected convex regions it can represent
  • A network with insufficient capacity cannot exactly model a function that requires a greater minimal number of convex hulls than the capacity of the network
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