Numpy로 NeuralNet 구현하기

Data 분리

import numpy as np
import pandas as pd
from matplotlib import pyplot as plt

data = pd.read_csv('/kaggle/input/digit-recognizer/train.csv')

data = np.array(data)
m, n = data.shape
np.random.shuffle(data) # shuffle before splitting into dev and training sets



data_train = data[:m].T
Y_train = data_train[0]
X_train = data_train[1:n]
X_train = X_train / 255.
_,m_train = X_train.shape

모델 파라미터

Forward propagation

$Z^{[1]} = W^{[1]} X + b^{[1]}$
$A^{[1]} = g_{\text{ReLU}}(Z^{[1]}))$
$Z^{[2]} = W^{[2]} A^{[1]} + b^{[2]}$
$A^{[2]} = g_{\text{softmax}}(Z^{[2]})$

Backward propagation

$dZ^{[2]} = A^{[2]} - Y$
$dW^{[2]} = \frac{1}{m} dZ^{[2]} A^{[1]T}$
$dB^{[2]} = \frac{1}{m} \Sigma {dZ^{[2]}}$
$dZ^{[1]} = W^{[2]T} dZ^{[2]} .* g^{[1]\prime} (z^{[1]})$
$dW^{[1]} = \frac{1}{m} dZ^{[1]} A^{[0]T}$
$dB^{[1]} = \frac{1}{m} \Sigma {dZ^{[1]}}$

Parameter updates

$W^{[2]} := W^{[2]} - \alpha dW^{[2]}$
$b^{[2]} := b^{[2]} - \alpha db^{[2]}$
$W^{[1]} := W^{[1]} - \alpha dW^{[1]}$
$b^{[1]} := b^{[1]} - \alpha db^{[1]}$

Feed forward

• $A^{[0]} = X$: 784 x m
• $Z^{[1]} \sim A^{[1]}$: 10 x m
• $W^{[1]}$: 10 x 784 (as $W^{[1]} A^{[0]} \sim Z^{[1]}$)
• $B^{[1]}$: 10 x 1
• $Z^{[2]} \sim A^{[2]}$: 10 x m
• $W^{[1]}$: 10 x 10 (as $W^{[2]} A^{[1]} \sim Z^{[2]}$)
• $B^{[2]}$: 10 x 1

Back propagation

• $dZ^{[2]}$: 10 x m ($~A^{[2]}$)
• $dW^{[2]}$: 10 x 10
• $dB^{[2]}$: 10 x 1
• $dZ^{[1]}$: 10 x m ($~A^{[1]}$)
• $dW^{[1]}$: 10 x 10
• $dB^{[1]}$: 10 x 1

함수정의

def init_params():
    W1 = np.random.rand(10, 784) - 0.5
    b1 = np.random.rand(10, 1) - 0.5
    W2 = np.random.rand(10, 10) - 0.5
    b2 = np.random.rand(10, 1) - 0.5
    return W1, b1, W2, b2

def ReLU(Z):
    return np.maximum(Z, 0)

def softmax(Z):
    A = np.exp(Z) / sum(np.exp(Z))
    return A
    
def forward_prop(W1, b1, W2, b2, X):
    Z1 = W1.dot(X) + b1
    A1 = ReLU(Z1)
    Z2 = W2.dot(A1) + b2
    A2 = softmax(Z2)
    return Z1, A1, Z2, A2

def ReLU_deriv(Z):
    return Z > 0

def one_hot(Y):
    one_hot_Y = np.zeros((Y.size, Y.max() + 1))
    one_hot_Y[np.arange(Y.size), Y] = 1
    one_hot_Y = one_hot_Y.T
    return one_hot_Y

def backward_prop(Z1, A1, Z2, A2, W1, W2, X, Y):
    one_hot_Y = one_hot(Y)
    dZ2 = A2 - one_hot_Y
    dW2 = 1 / m * dZ2.dot(A1.T)
    db2 = 1 / m * np.sum(dZ2)
    dZ1 = W2.T.dot(dZ2) * ReLU_deriv(Z1)
    dW1 = 1 / m * dZ1.dot(X.T)
    db1 = 1 / m * np.sum(dZ1)
    return dW1, db1, dW2, db2

def update_params(W1, b1, W2, b2, dW1, db1, dW2, db2, alpha):
    W1 = W1 - alpha * dW1
    b1 = b1 - alpha * db1    
    W2 = W2 - alpha * dW2  
    b2 = b2 - alpha * db2    
    return W1, b1, W2, b2
    
def get_predictions(A2):
    return np.argmax(A2, 0)

def get_accuracy(predictions, Y):
    print(predictions, Y)
    return np.sum(predictions == Y) / Y.size

def gradient_descent(X, Y, alpha, iterations):
    W1, b1, W2, b2 = init_params()
    for i in range(iterations):
        Z1, A1, Z2, A2 = forward_prop(W1, b1, W2, b2, X)
        dW1, db1, dW2, db2 = backward_prop(Z1, A1, Z2, A2, W1, W2, X, Y)
        W1, b1, W2, b2 = update_params(W1, b1, W2, b2, dW1, db1, dW2, db2, alpha)
        if i % 10 == 0:
            print("Iteration: ", i)
            predictions = get_predictions(A2)
            print(get_accuracy(predictions, Y))
    return W1, b1, W2, b2

학습 & 결과

W1, b1, W2, b2 = gradient_descent(X_train, Y_train, 0.10, 500)
Iteration:  0
[2 6 4 ... 0 3 0] [1 3 8 ... 3 7 4]
0.10939024390243902
Iteration:  10
[2 5 7 ... 1 3 0] [1 3 8 ... 3 7 4]
0.16495121951219513
Iteration:  20
[2 5 7 ... 3 3 0] [1 3 8 ... 3 7 4]
0.20821951219512194
Iteration:  30
[2 3 7 ... 3 3 0] [1 3 8 ... 3 7 4]
0.24424390243902439
Iteration:  40
[2 3 7 ... 1 3 0] [1 3 8 ... 3 7 4]
0.281780487804878
Iteration:  50
[2 3 7 ... 3 3 0] [1 3 8 ... 3 7 4]
0.3227317073170732
Iteration:  60
[2 3 7 ... 3 9 0] [1 3 8 ... 3 7 4]
0.3648048780487805
Iteration:  70
[2 3 7 ... 3 9 0] [1 3 8 ... 3 7 4]
0.40904878048780485
Iteration:  80
[2 3 7 ... 3 7 0] [1 3 8 ... 3 7 4]
0.450609756097561
Iteration:  90
[2 3 7 ... 3 7 0] [1 3 8 ... 3 7 4]
0.4898292682926829
Iteration:  100
[2 8 7 ... 3 7 0] [1 3 8 ... 3 7 4]
0.5265121951219512
Iteration:  110
[2 8 7 ... 3 7 0] [1 3 8 ... 3 7 4]
0.5636097560975609
Iteration:  120
[2 8 7 ... 3 7 0] [1 3 8 ... 3 7 4]
0.5929268292682927
Iteration:  130
[2 8 7 ... 3 7 0] [1 3 8 ... 3 7 4]
0.6162682926829268
Iteration:  140
[2 8 7 ... 3 7 0] [1 3 8 ... 3 7 4]
0.6360975609756098
Iteration:  150
[2 8 7 ... 3 7 0] [1 3 8 ... 3 7 4]
0.6531219512195122
Iteration:  160
[2 3 7 ... 3 7 0] [1 3 8 ... 3 7 4]
0.6678536585365854
Iteration:  170
[2 3 7 ... 3 7 4] [1 3 8 ... 3 7 4]
0.6811707317073171
Iteration:  180
[2 3 7 ... 8 7 4] [1 3 8 ... 3 7 4]
0.6940487804878048
Iteration:  190
[2 3 7 ... 8 7 4] [1 3 8 ... 3 7 4]
0.7065609756097561
Iteration:  200
[2 3 7 ... 8 7 4] [1 3 8 ... 3 7 4]
0.7169756097560975
Iteration:  210
[2 3 7 ... 8 7 4] [1 3 8 ... 3 7 4]
0.7270243902439024
Iteration:  220
[2 3 7 ... 8 7 4] [1 3 8 ... 3 7 4]
0.7370487804878049
Iteration:  230
[2 3 7 ... 8 7 4] [1 3 8 ... 3 7 4]
0.745780487804878
Iteration:  240
[2 3 7 ... 8 7 4] [1 3 8 ... 3 7 4]
0.7534634146341463
Iteration:  250
[2 3 7 ... 8 7 4] [1 3 8 ... 3 7 4]
0.7608292682926829
Iteration:  260
[2 3 7 ... 8 7 4] [1 3 8 ... 3 7 4]
0.7681951219512195
Iteration:  270
[2 3 7 ... 8 7 4] [1 3 8 ... 3 7 4]
0.7747073170731708
Iteration:  280
[2 3 7 ... 8 7 4] [1 3 8 ... 3 7 4]
0.7802926829268293
Iteration:  290
[2 3 7 ... 8 7 4] [1 3 8 ... 3 7 4]
0.7852439024390244
Iteration:  300
[2 3 7 ... 8 7 4] [1 3 8 ... 3 7 4]
0.7903170731707317
Iteration:  310
[2 3 7 ... 8 7 4] [1 3 8 ... 3 7 4]
0.795219512195122
Iteration:  320
[2 3 5 ... 8 7 4] [1 3 8 ... 3 7 4]
0.7995365853658537
Iteration:  330
[2 3 5 ... 8 7 4] [1 3 8 ... 3 7 4]
0.8032439024390244
Iteration:  340
[2 3 5 ... 8 7 4] [1 3 8 ... 3 7 4]
0.8069756097560976
Iteration:  350
[2 3 5 ... 8 7 4] [1 3 8 ... 3 7 4]
0.8109024390243903
Iteration:  360
[2 3 5 ... 8 7 4] [1 3 8 ... 3 7 4]
0.8143170731707317
Iteration:  370
[2 3 5 ... 8 7 4] [1 3 8 ... 3 7 4]
0.8172195121951219
Iteration:  380
[2 3 5 ... 8 7 4] [1 3 8 ... 3 7 4]
0.8200975609756097
Iteration:  390
[6 3 5 ... 8 7 4] [1 3 8 ... 3 7 4]
0.8228048780487804
Iteration:  400
[6 3 5 ... 8 7 4] [1 3 8 ... 3 7 4]
0.8251219512195122
Iteration:  410
[6 3 5 ... 8 7 4] [1 3 8 ... 3 7 4]
0.8273170731707317
Iteration:  420
[6 3 5 ... 8 7 4] [1 3 8 ... 3 7 4]
0.8297317073170731
Iteration:  430
[6 3 5 ... 8 7 4] [1 3 8 ... 3 7 4]
0.8323414634146341
Iteration:  440
[6 3 5 ... 8 7 4] [1 3 8 ... 3 7 4]
0.8342682926829268
Iteration:  450
[6 3 5 ... 8 7 4] [1 3 8 ... 3 7 4]
0.8360975609756097
Iteration:  460
[6 3 5 ... 8 7 4] [1 3 8 ... 3 7 4]
0.8379268292682926
Iteration:  470
[6 3 5 ... 8 7 4] [1 3 8 ... 3 7 4]
0.8395365853658536
Iteration:  480
[6 3 5 ... 8 7 4] [1 3 8 ... 3 7 4]
0.8408536585365853
Iteration:  490
[6 3 5 ... 8 7 4] [1 3 8 ... 3 7 4]
0.8427073170731707 	########################약 85%의 정확도

def make_predictions(X, W1, b1, W2, b2):
    _, _, _, A2 = forward_prop(W1, b1, W2, b2, X)
    predictions = get_predictions(A2)
    return predictions

def test_prediction(index, W1, b1, W2, b2):
    current_image = X_train[:, index, None]
    prediction = make_predictions(X_train[:, index, None], W1, b1, W2, b2)
    label = Y_train[index]
    print("Prediction: ", prediction)
    print("Label: ", label)
    
    current_image = current_image.reshape((28, 28)) * 255
    plt.gray()
    plt.imshow(current_image, interpolation='nearest')
    plt.show()
   
############################################################
test_prediction(100, W1, b1, W2, b2)
test_prediction(101, W1, b1, W2, b2)
test_prediction(102, W1, b1, W2, b2)
test_prediction(103, W1, b1, W2, b2)  

Prediction: [1]
Label: 1

Prediction: [1]
Label: 8

Prediction: [1]
Label: 1

Prediction: [9]
Label: 9