LEE CHANWOO

📝 Blog 2024 version 1.7.0
🔖 NOV 26, 2024

5 분 소요

Multi-Layer Perceptiron, MLP

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  • MLP and backpropagation algorithm which is used to train it
  • MLP used to describe any general feedforward (no recurrent connections) network


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  • 3 layer (no. of layers of adaptive weights), 3층 신경망
  • 1st question:
    • 은닉층의 역할?
      • 특징추출기
    • 단층 신경망의 문제점?


XOR problem

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  • XOR (exclusive OR) problem
    • 0+0=0
    • 1+1=2=0 mod 2
    • 1+0=1
    • 0+1=1
  • Perceptron does not work here
  • Single layer generates a linear decision boundary


  • Minsky & Papert (1969) offered solution to XOR problem by combining perceptron unit responses using a second layer of units

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  • $out = hard lim(w_1 x_1 + w_2 x_2 - \theta)$


Three Layer Networks

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Properties of architecture

  • 층내에서 연결이 없음
  • 입력층과 출력층의 직접연결이 없음
  • 인접층간에는 완전연결
  • 출력 unit개수, 입력 unit 개수
  • 은닉층의 유닛의 개수는 입력유닛 혹은 출력 유닛의 개수보다 작거나, 많을수 있다.

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Decision Boundary

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  • 2층은 지역적 지식을 추출하고, 3층은 전역적인 지식을 추출
  • 은닉층에 sigmoidal 구동함수를 사용하면, 2층 신경망은 어떤 함수라도 근사화 할 수 있음. : Universal Approximation


Backpropagation Learning

  • 단층 퍼셉트론에서 가중치를 찾기 위해, 손실함수를 경사하강법을 적용 :
    • $\Delta w_{ji} = (t_j - y_j) x_i$
  • 입력유닛 $i$ 에서 출력 유닛 $j$로의 가중치 ($w_{ji}$) 의 수정량은 입력과 $j$ 출력 유닛의 에러에만 의존하는 지역적 특성

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  • 에러는 3째층에서만 계산됨. 첫째층과 2째층의 가중치는?
  • 처음 2개층에는 직접적인 에러가 없음.


  • Credit assignment problem
    • Problem of assigning ‘credit’ or ‘blame’ to individual elements involved in forming overall response of a learning system (hidden units)
    • In neural networks, problem relates to deciding which weights should be altered, by how much and in which direction


  • 초기층의 가중치가 출력, 따라서 에러에 얼마나 기여하는지를 결정해야함
  • 즉, 가중치 wij 가 에러에 어떤 영향을 끼치는지 구하고자 함. 다음 값을 구하고자 함.

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  • Backpropagation learning algorithm ‘BP’

  • Solution to credit assignment problem in MLP. Rumelhart, Hinton and Williams (1986)

  • BP 는 두 단계:
    • 순방향 패스: 순방향으로 각 유닛의 입력값 계산, 출력 계산을 반복하여 최종 출력값 구함.
    • 역방향 패스: 출력 유닛의 에러신호를 계산한후 역방향으로 에러를 전파함. (에러는 실제값과 목표치와의 차이)


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  • 2층으로 설명함. 쉽게 다층으로 확장 가능함.

  • $z_i(t) = g( \sum_j v_{ij}(t) x_j(t) )$ at time t
    • $= g ( u_i(t) )$
  • $y_i (t) = g( \sum_j w_{ij}(t)z_j(t) )$ at time t
    • $= g ( a_i(t) )$
  • a/u activation(total 입력)
  • g 구동함수(activation function)
  • 바이어스는 추가 가중치로 둠

Forward pass

  • Weights are fixed during forward and backward pass at time t

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Backward Pass

Will use a sum of squares error measure. For each training pattern we have:


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where dk is the target value for dimension k. We want to know how to modify weights in order to decrease E. Use gradient descent ie


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both for hidden units and output units


  • The partial derivative can be rewritten as product of two terms using chain rule for partial differentiation

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both for hidden units and output units

  • Term A
    • i 유닛의 전체입력 변화에 따라 에러가 어떻게 변하는가? $\Delta i$
  • Term B
    • 가중치 w 의 변화에 따라 i 유닛의 입력은 어떻게 변하는가? $z_j$


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Activation Functions

  • How does the activation function affect the changes?

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  • we need to compute the derivative of activation function g
  • to find derivative the activation function must be smooth (differentiable)


  • Sigmoidal (logistic) function-common in MLP

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  • Derivative of sigmoidal function is

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  • Derivative of sigmoidal function has max at 𝑎𝑖(𝑡)= 0, is symmetric about this point falling to zero as sigmoid approaches extreme values


  • 가중치의 수정값은 구동함수의 미분에 비례함.

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  • 가중치 수정은 유닛의 전체입력이 중간값(0부근)일 때 가장 크게 발생. 입력이 아주 크거나,아주 작을 때에는 수정치는 거의 0임. 유닛이 포화상태. …다층일 때 그래디언트 소멸(gradient vanish)문제


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Batch Learning, Online learning

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  • Batch learning :
    • 각 훈련데이터에 대한 계산된 가중치 수정치를 누적함
    • 전체 훈련데이터에 대해 누적된 값을 한번에 수정함- –epoch
    • 수정 횟수 적음
  • Online learning :
    • 가중치를 각 훈련데이터마다 수정함
    • 소요 메모리 공간 적음
    • 학습률은 작은 값으로 둠
    • 데이터는 랜덤한 순서로 입력함 - stochastic gradient descent
    • 랜덤순서…지역적 최소값을 회피하는데 도움


Activation function of output unit

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Learning Example

  • Once weight changes are computed for all units, weights are updated at the same time (bias included as weights here). An example:

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  • Use identity activation function (ie g(a) = a)


  • All biases set to 1. Will not draw them for clarity.
  • Learning rate $\eta$ = 0.1

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Have input [0 1] with target [1 0].


  • Forward pass. Calculate $1^{st}$ layer activations:

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  • $u_1 = -1 \times 0 + 0 \times 1 + 1 = 1$
  • $u_2 = 0 \times 0 + 1 \times 1 + 1 = 2$


  • Calculate first layer outputs by passing activations thru activation functions

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  • $z_1 = g(u_1) = 1$
  • $z_2 = g(u_2) = 2$


  • Calculate $2^{nd}$ layer outputs (weighted sum thru activation functions):

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  • $y_1 = a_1 = 1 \times 1 + 0 \times 2 + 1 = 2$
  • $y_2 = a_2 = -1 \times 1 + 1 \times 2 + 1 = 2$


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  • So
    • $\Delta_1 = (d_1 - y_1) = 1 – 2 = -1$
    • $\Delta_2 = (d_2 - y_2) = 0 – 2 = -2$


  • Calculate weight changes for $1^{st}$ layer (cf perceptron learning):

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  • $\delta_1 = - 1 + 2 = 1$
  • $\delta_2 = 0 – 2 = -2$


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Finally change weights:

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  • Note that the weights multiplied by the zero input are unchanged as they do not contribute to the error We have also changed biases (not shown)


  • Now go forward again (would normally use a new input vector):

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  • Now go forward again (would normally use a new input vector):

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  • Outputs now closer to target value [1, 0]


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Selecting initial weight values

  • Choice of initial weight values is important as this decides starting position in weight space. That is, how far away from global minimum
  • Select weight values randomly from uniform probability distribution


Momentum

  • 수렴속도를 높이면서 불안정성을 줄이는 기법
  • 가중치 갱신식에 이전 가중치 수정치의 비례값을 더함
  • 수정된 가중치 갱신식

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  • $\alpha$ is momentum constant and controls how much notice is taken of recent history

  • Effect of momentum term

    • 가중치의 수정치가 이전 수정치와 같은 부호를 가지면, 관성항은 수정을 많이 하여 수렴속도를 높임.
    • 만약 수정치가 이전의 수정치와 반대부호를 가지면, 관성항은 수정치를 줄여서 속도를 늦추어 진동을 방지한다. (안정화)
    • 지역적 최소치를 회피하도록 도움


Adaptive Learning Rate

  • Learning rate $\eta$
    • mostly less than or equal to 0.2
    • can be made adaptive for faster convergence
    • kept large when learning takes place and decreased when learning slows down


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Overtraining

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  • where
    • n = number of training patterns,
    • m = number of output units

  • Could stop training when rate of change of E is small, suggesting convergence

  • However, aim is for new patterns to be classified correctly


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  • Typically, though error on training set will decrease as training continues generalisation error (error on unseen data) hitts a minimum then increases (cf model complexity etc)
  • Therefore want more complex stopping criterion


  • Cross-validation
    • Method for evaluating generalisation performance of networks in order to determine which is best using of available data
  • Hold-out method
    • Simplest method when data is not scare
  • Divide available data into sets
    • Training data set
      • used to obtain weight and bias values during network training
    • Validation data
      • used to periodically test ability of network to generalize

      • suggest ‘best’ network based on smallest error

  • Cf . Test data


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Weka

Choose classify-functions-multilayer perceptron

  1. hidden layer 의 노드 수의 설정방법
    popup 창의 choose 난의 multilayer perceptron을 클릭
    hiddenLayers 에 은닉층의 노드수 입력
    예 2,4 입력
    1째 은닉층의 노드수 2개 2째 은닉층의 노드수 4개
  2. choose 난의 multilayer perceptron을 클릭 popup 창의 GUI 난을 true 로 하면 MLP 구조가 보임

MLP regressor (cpu performance)

MLP classifier(iris)

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