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Regression and Classification SC4001 – Tutorial 2
࠵? = ࠵? ! , ࠵? " , ࠵? # ࠵? 0.09, −0.44, −0.15 −2.57 0.69, −0.99, −0.76 −2.97 0.34, 0.65, −0.73 0.96 0.15, , 0.78, −0.58 1.04 −0.63, −0.78, −0.56 −3.21 0.96, 0.62, −0.66 1.05 0.63, −0.45, −0.14 −2.39 0.88, 0.64, −0.33 0.66
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initial weights and biases: ࠵? = 0.77 0.02 0.63 , ࠵? = 0.0 Learning factor ࠵? = 0.01 ࠵? ! ࠵? " ࠵? # ࠵? +1 ࠵? = ࠵? ࠵? = ࠵?
Given a training dataset ࠵? $ , ࠵? $ $%! & Set learning parameter ࠵? Initialize ࠵? and ࠵? Repeat until convergence: For every training pattern ࠵? $ , ࠵? $ : ࠵? $ = ࠵? $ ’ ࠵? + ࠵? ࠵? ← ࠵? + ࠵? ࠵? $ − ࠵? $ ࠵? $ ࠵? ← ࠵? + ࠵? ࠵? $ − ࠵? $ (a) SGD for the linear neuron :
SGD: learning factor ࠵? = 0.01 Shuffle the inputs. Epoch 1 ࠵? = 1 Apply ࠵? $ = 0.34 0.65 −0.73 , ࠵? $ = 0.96 ࠵? $ = ࠵? ࠵? & ࠵? + ࠵? = 0.34 0.65 −0.73 0.77 0.02 0.63 + 0.0 = −0.19 ࠵?. ࠵?. = ࠵? $ − ࠵? $ " = 0.96 + 0.19 " = 1.31 ࠵? ← ࠵? + ࠵? ࠵? $ − ࠵? $ ࠵? ࠵? = 0.77 0.02 0.63 + 0.01× 0.96 + 0.19 0.34 0.65 −0.73 = 0.78 0.03 0.62 ࠵? ← ࠵? + ࠵? ࠵? $ − ࠵? $ = 0.0 + 0.01× 0.96 + 0.19 = 0.01
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࠵? = 2 Apply ࠵? ࠵? = 0.63 −0.45 −0.14 , ࠵? $ = −2.39 ࠵? $ = ࠵? ࠵? & ࠵? + ࠵? = 0.63 −0.45 −0.14 0.78 0.03 0.63 + 0.01 = 0.4 ࠵?. ࠵?. = ࠵? $ − ࠵? $ " = −2.39 − 0.4 " = 7.78 ࠵? ← ࠵? + ࠵? ࠵? $ − ࠵? $ ࠵? ࠵? = 0.78 0.03 0.63 + 0.01× −2.39 − 0.4 0.63 −0.45 −0.14 = 0.76 0.04 0.63 ࠵? ← ࠵? + ࠵? ࠵? $ − ࠵? $ = 0.01 + 0.01× −2.39 − 0.4 = −0.02 Continue apply other patterns …. Then continue epochs 2, 3, …… until convergence.
Epoch 1 ࠵? ࠵? ࠵? s.e. ࠵? ࠵? 0.34 0.65 −0.73 0.96 -0.19 1.31 0.78 0.03 0.63 0.01 0.63 −0.45 −0.14 -2.39 0.4 7.78 0.76 0.04 0.63 -0.02 0.88 0.64 −0.33 0.66 0.47 0.04 0.76 0.04 0.63 -0.01 0.96 0.62 −0.66 1.05 0.33 0.52 0.77 0.05 0.62 -0.01 0.09 −0.44 −0.15 -2.57 -0.05 6.34 0.76 0.06 0.63 -0.03 0.69 −0.99 −0.76 -2.97 -0.04 8.59 0.74 0.09 0.65 -0.06 −0.63 −0.78 −0.56 -3.21 -0.96 5.05 0.76 0.10 0.66 -0.08 0.15 0.78 −0.58 1.04 -0.27 1.73 0.76 0.11 0.65 -0.07 m.s.e = 3.92
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At convergence: ࠵? = 0.37 2.57 −0.21 , ࠵? = −1.17 mse = 0.054 If ࠵? = ࠵? ! , ࠵? " , ࠵? # ’ , the learned function by the linear neuron: ࠵? = ࠵? ࠵? ࠵? + ࠵? ࠵? = 0.37 2.57 −0.21 ࠵? ! ࠵? " ࠵? # − 1.17 ࠵? = 0.37࠵? ! + 2.57࠵? " − 0.21࠵? # − 1.17
x: [-0.63 -0.78 -0.56], d: -3.21, y: -3.28035 x: [ 0.96 0.62 -0.66], d: 1.05, y: 0.919454 x: [ 0.09 -0.44 -0.15], d: -2.57, y: -2.22926 x: [ 0.88 0.64 -0.33], d: 0.66, y: 0.871378 x: [ 0.34 0.65 -0.73], d: 0.96, y: 0.782616 x: [ 0.15 0.78 -0.58], d: 1.04, y: 1.01435 x: [ 0.69 -0.99 -0.76], d: -2.97, y: -3.29005 x: [ 0.63 -0.45 -0.14], d: -2.39, y: -2.0579 Predicted values: ࠵? = 0.37࠵? ! + 2.57࠵? " − 0.21࠵? # − 1.17
(b) GD for a linear neuron Given a training dataset ࠵?, ࠵? Set the learning parameter ࠵? Initialize ࠵? and ࠵? Repeat until convergence: ࠵? = ࠵?࠵? + ࠵?࠵? & ࠵? ← ࠵? + ࠵?࠵? ’ ࠵? − ࠵? ࠵? ← ࠵? + ࠵?࠵? & ’ ࠵? − ࠵?
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࠵? = 0.09 −0.44 −0.15 0.69 −0.99 −0.76 0.34 0.65 −0.73 0.15 0.78 −0.58 −0.63 −0.78 −0.56 0.96 0.62 −0.66 0.63 −0.45 −0.14 0.88 0.64 −0.33 , ࠵? = −2.57 −2.97 0.96 1.04 −3.21 1.05 −2.39 0.66 GD for a linear neuron initial weights and biases: ࠵? = 0.77 0.02 0.63 , ࠵? = 0.0 Learning factor ࠵? = 0.01
Output ࠵? = ࠵?࠵? + ࠵?࠵? ’ = 0.09 −0.44 −0.15 0.69 −0.99 −0.76 0.34 0.65 −0.73 0.15 0.78 −0.58 −0.63 −0.78 −0.56 0.96 0.62 −0.66 0.63 −0.45 −0.14 0.88 0.64 −0.33 0.77 0.02 0.63 + 0.0 1 1 1 1 1 1 1 1 = −0.03 0.03 −0.18 −0.24 −0.85 0.34 0.39 0.48 m.s.e. = ! ( ∑ $)! ( ࠵? $ − ࠵? $ " = 1 8 −2.57 + 0.03 " + −2.97 − 0.03 " + ⋯ ⋯ + 0.66 − 0.48 " = 4.02 ࠵? = −2.57 −2.97 0.96 1.04 −3.21 1.05 −2.39 0.66
࠵? = ࠵? + ࠵?࠵? & ࠵? − ࠵? = 0.77 0.02 0.63 + 0.01× 0.09 0.69 0.34 0.15 −0.63 0.96 0.63 0.88 −0.44 −0.99 0.65 0.78 −0.78 0.62 −0.45 0.64 −0.15 −0.76 −0.73 −0.58 −0.56 −0.66 −0.14 −0.33 −2.57 −2.97 0.96 1.04 −3.21 1.05 −2.39 0.66 − −0.03 0.03 −0.18 −0.24 −0.85 0.34 0.39 0.48 = 0.76 0.11 0.65 ࠵? = ࠵? + ࠵?࠵? ’ & ࠵? − ࠵? = 0.0 + 0.01× 1 1 1 1 1 1 1 1 −2.57 −2.97 0.96 1.04 −3.21 1.05 −2.39 0.66 − −0.03 0.03 −0.18 −0.24 −0.85 0.34 0.39 0.48 = −0.07
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epoch ࠵? mse ࠵? ࠵? 2 −0.15 −0.16 −0.22 −0.25 −1.01 0.29 0.26 0.45 3.66 0.75 0.21 0.67 - 0.14 200 −2.23 −3.29 0.78 1.01 −3.28 0.92 −2.06 0.87 0.05 0.368 2.56 −0.21 - 1.164
At convergence: ࠵? = 0.37 2.57 −0.21 , ࠵? = −1.16 mse = 0.054 If ࠵? = ࠵? ! , ࠵? " , ࠵? # ’ , the learned function by the linear neuron: ࠵? = ࠵? ࠵? ࠵? + ࠵? ࠵? = 0.37 2.57 −0.21 ࠵? ! ࠵? " ࠵? # − 1.16 ࠵? = 0.37࠵? ! + 2.57࠵? " − 0.21࠵? # − 1.16
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࠵? ࠵? ࠵? (SGD) ࠵? (GD) 0.09, −0.44, −0.15 −2.57 −2.23 −2.23 0.69, −0.99, −0.76 −2.97 −3.29 −3.29 0.34, 0.65, −0.73 0.96 0.78 0.78 0.15, , 0.78, −0.58 1.04 1.01 1.01 −0.63, −0.78, −0.56 −3.21 −3.28 −3.28 0.96, 0.62, −0.66 1.05 0.92 0.92 0.63, −0.45, −0.14 −2.39 −2.06 −2.06 0.88, 0.64, −0.33 0.66 0.87 0.88 m.s.e. 0.055 0.054 ࠵? 0.369 2.566 −0.212 0.368 2.567 −0.207 ࠵? −1.165 - 1.163 ࠵? 0.37࠵? ! + 2.57࠵? " − 0.21࠵? # − 1.1 0.37࠵? ! + 2.57࠵? " − 0.21࠵? # − 1.16 After learning …
࠵? ! = 5 1 , ࠵? " = 7 3 , ࠵? # = 3 2 , ࠵? 9 = 5 4 → class 1 ࠵? : = 0 0 , ࠵? ; = −1 −3 , ࠵? < = −2 3 , ࠵? = = −3 0 → class 2 Center of class-1: ࠵? ! = ! * ࠵? ! + ࠵? " + ࠵? # + ࠵? * = 5.0 2.5 For class-2: ࠵? " = ! * ࠵? + + ࠵? , + ࠵? - + ࠵? ( = −1.5 0.0 The two classes are linearly separable!
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The vector connecting two centroids = ࠵? ! − ࠵? " = 6.5 2.5 (1) The middle point connecting two centroids = ! " ࠵? ! + ࠵? " = ! " 5.0 2.5 + −1.5 0.0 = 1.75 1.25 If any point ࠵? ! ࠵? " on the boundary line, the vector connecting that point to the middle point = ࠵? ! ࠵? " − 1.75 1.25 = ࠵? ! − 1.75 ࠵? " − 1.25 (2) Since the vectors (1) and (2) are normal, their inner product should be zero. 6.5 2.5 ࠵? ! − 1.75 ࠵? " − 1.25 = 0 6.5 ࠵? ! − 1.75 + 2.5 ࠵? " − 1.25 = 0 6.5࠵? ! + 2.5࠵? " − 14.5 = 0
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+1 ࠵? ! ࠵? " ࠵? 6.5 2.5 -14.5 ࠵? = 6.5࠵? ! + 2.5࠵? " − 14.5 = 0 Perceptron: Weights: ࠵? = 6.5 2.5 Bias: ࠵? = −14.5
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࠵? = 6.5࠵? ! + 2.5࠵? " − 14.5 ࠵? = ࠵? & ࠵? + ࠵? = 6.5 2.5 ࠵? ! ࠵? " − 14.5 ࠵? 5 1 7 3 3 2 5 4 0 0 −1 −3 −2 3 −3 0 ࠵? 20.5 38.5 10.0 28.0 -14.5 -28.5 -20.0 -34.0 class 1 1 1 1 2 2 2 2 ࠵? > 0.0 → ࠵?࠵?࠵?࠵?࠵? 1 ࠵? ≤ 0.0 → ࠵?࠵?࠵?࠵?࠵? 2 Training patterns Testing patterns: ࠵? = 4 2 , ࠵? = 16.5 > 0 → ࠵?࠵?࠵?࠵?࠵? 1 ࠵? = 0 5 , ࠵? = −2 ≤ 0 → ࠵?࠵?࠵?࠵?࠵? 2 ࠵? = 36/13 0 , ࠵? = 3.5 > 0 → ࠵?࠵?࠵?࠵?࠵? 1 Perfectly classified!
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28 from sklearn.datasets import make_blobs # define dataset X, y = make_blobs (n_samples=100, n_features=3, centers=2, cluster_std=5, random_state=1) # print first few examples for i in range(5): print(X[i], y[i]) [ 5.82704597 -8.69737967 -14.86660705] 1 [ -5.95713961 6.15921976 -16.55912956] 0 [-7.16265579 10.13010842 -5.4897589 ] 0 [-5.19652244 -6.84653722 -9.28479932] 1 [ 0.9050892 2.91602569 -7.55512177] 0
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# create a class for a logistic neuron class Logistic(): def __init__(self): self.w = torch.tensor(0.05*np.random.rand(3), dtype=torch.double, requires_grad= True ) self.b = torch.tensor(0., dtype=torch.double, requires_grad= True ) def __call__(self, x): u = torch.matmul(torch.tensor(x), self.w) + self.b logits = torch.sigmoid(u) return logits import torch import numpy as np
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31 import torch.nn as nn loss = nn.BCELoss () # create a logistic neuron object model = Logistic() # one iteration of training def train(model, inputs, targets, learning_rate): logits = model(inputs) loss_ = loss(logits, torch.tensor(targets, dtype=torch.double)) err = torch.sum(torch.not_equal(logits > 0.5, torch.tensor(targets))) loss_. backward () with torch.no_grad(): model.w -= learning_rate * model.w.grad model.b -= learning_rate * model.b.grad model.w.grad = None model.b.grad = None return loss_, err
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# training begins idx = np.arange(100) entropy, err = [], [] for epoch in range(100): np.random.shuffle(idx) entropy_, err_ = train (model, X[idx], y[idx], lr) entropy.append(entropy_.detach().numpy()), err.append(err_.detach().numpy())
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Learning curves
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At convergence : w: [-0.07863051 -0.38637461 0.04367386], b: -0.0053 entropy: 0.30624, error: 14 Decision boundary: ࠵? = ࠵? ’ ࠵? + ࠵? = 0 −0.079 −0.386 0.044 ࠵? ! ࠵? " ࠵? # − 0.005 = 0 −0.079࠵? ! − 0.386࠵? " + 0.044࠵? # + 0.005 = 0
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࠵? = ࠵? ࠵?, ࠵? = 1.5 + 3.3࠵? − 2.5࠵? + 1.2࠵?࠵? for every 0 ≤ ࠵?, ࠵? ≤ 1 Training dataset was created by randomly sampling the input feature space and computing the corresponding labels.
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For perceptron , the activation function is a sigmoidal and the range of output should be known. Let: ࠵? = ࠵? ࠵?, ࠵? = 1.5 + 3.3࠵? − 2.5࠵? + 1.2࠵?࠵? where 0 ≤ ࠵?, ࠵? ≤ 1 ࠵? ࠵? +1 ࠵? Differentiating the function to find the maximum and minimum points: . ࠵?࠵? ࠵?࠵? = 3.3 + 1.2࠵? = 0 → ࠵? = −2.75 ࠵?࠵? ࠵?࠵? = −2.5 + 1.2࠵? = 0 → ࠵? = 2.08 That is, maximum/minimum occurs outside the given region. The maximum/minimum occur at boundary points for the given input region.
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For the perceptron, activation function: ࠵? ࠵? = 5.8 1 + ࠵? ./ − 1.0 (࠵?, ࠵?) ࠵?, ࠵? ࠵?, ࠵? ࠵?, ࠵? ࠵?, ࠵? ࠵? 1.5 4.8 -1.0 3.5 Therefore, ࠵? ∈ −1.0, 4.8
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# a class for the preceptron class Perceptron(): def __init__(self): self.w = torch.tensor(np.random.rand(2,1), dtype=torch.double, requires_grad=True) self.b = torch.tensor(0., dtype=torch.double, requires_grad=True) def __call__(self, x): u = torch.matmul(torch.tensor(x), self.w) + self.b y = 5.8*torch.sigmoid(u)-1.0 return y # mean squared error as the loss function def loss_fn(y_pred, d): return torch.mean(torch.square(y_pred - d)) # create a linear neuron model = Perceptron()
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# training idx = np.arange(no_data) for epoch in range(no_epochs): np.random.shuffle(idx) XX, YY = X[idx], Y[idx] y_ = model(XX) loss_ = loss_fn(y_, torch.tensor(YY)) loss_.backward() with torch.no_grad(): model.w -= lr * model.w.grad model.b -= lr * model.b.grad model.w.grad = None model.b.grad = None
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At convergence: ࠵? = 2.95 −1.91 , ࠵? = −0.48 mse = 0.014 If ࠵? = ࠵?, ࠵? ’ ࠵? = ࠵? ࠵? ࠵? + ࠵? ࠵? = 2.95 −1.91 ࠵? ࠵? − 0.48 ࠵? = 2.95࠵? − 1.91࠵? − 0.48 The learned function by the perceptron: ࠵? = 5.8 1 + ࠵? >? − 1.0 ࠵? = 5.8 1 + ࠵? @.9=>".A:BC!.A!D − 1.0
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Linear neuron learns a linear function. The above equation can be written as a linear equation: ࠵? = 1.5 + 3.3࠵? − 2.5࠵? + 1.2࠵?࠵? ࠵? = 1.5 + 3.3࠵? ! − 2.5࠵? " + 1.2࠵? # where the linear neuron receives 3 inputs: ࠵? ! = ࠵? , ࠵? " = ࠵? , and ࠵? # = ࠵?࠵?. ࠵? ࠵? ࠵?࠵? ࠵? +1 ࠵? ࠵? = ࠵?
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At convergence, Weights ࠵? = 3.40 −2.41 1.02 and bias ࠵? = 1.45 Mean square error = 1.8 x 10 -4 The function learned by the linear neuron: ࠵? = ࠵? ! ࠵? ! + ࠵? " ࠵? " + ࠵? # ࠵? # + ࠵? ࠵? = 3.40࠵? − 2.41࠵? + 1.02࠵?࠵? + 1.45
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Linear Neuron Perceptron ࠵? 3.40 −2.41 1.02 2.95 −1.91 ࠵? 1.45 -0.48 m.s.e. 1. 8x 10 -4 0.014 function ࠵? = 3.4࠵? − 2.41࠵? + 1.02࠵?࠵? + 1.45 ࠵? = 5.8 1 + ࠵? 0.*(.".2+34!.2!5 − 1.0 ࠵? = 1.5 + 3.3࠵? − 2.5࠵? + 0.2࠵?࠵? for 0 ≤ ࠵?, ࠵? ≤ 1
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• ࠵? = 1.5 + 3.3࠵? − 2.5࠵? + 0.2࠵?࠵? for 0 ≤ ࠵?, ࠵? ≤ 1 ࠵? = 3.4࠵? − 2.41࠵? + 1.02࠵?࠵? + 1.45 ࠵? = 5.8 1 + ࠵? !.#$%&.’()*+.’+, − 1.0
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