前言

教你怎么只用 numpy 和数学方法构建一个神经网络，而不是使用 Tensorflow 或 Pytorch。

相关资源

• Building a neural network FROM SCRATCH (no Tensorflow/Pytorch, just numpy & math) - YouTube

  • 这个视频有点坑，照着视频写代码会有很多错误😅

• Simple MNIST NN from scratch (numpy, no TF/Keras) | Kaggle

  • 原作者的笔记，还比较靠谱

内容

导入相关库

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

读取数据集

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

data.shape

(42000, 785)

数组形状:$(42000, 785)$, 说明:

• $42000$ 行, 表示这个数据集有 $42000$ 张图片

• $785$ 列, 表示数据集中每张图片大小为 $28 \times 28$, 外带 $1$ 个标签, $28 \times 28 + 1 = 785$

将数据集分为训练集和测试集

data = np.array(data)  # 将数据集从 pd.DataFrame 转为 np.array
m, n = data.shape
np.random.shuffle(data)  # shuffle 直接在原来的数组上进行操作，改变原来数组的顺序，无返回值

# 验证集
data_dev = data[0:1000].T  # 前 1000 个
Y_dev = data_dev[0]
X_dev = data_dev[1:n]
X_dev = X_dev / 255.  # 将图像各像素点的灰度值映射到 0 到 1 之间的浮点数，使之后的 exp 运算不会溢出 

# 训练集
data_train = data[1000:m].T
Y_train = data_train[0]
X_train = data_train[1:n]
X_train = X_train / 255.
_,m_train = X_train.shape

定义相关函数

​ Our NN will have a simple two-layer architecture. Input layer $a^{[0]}$ will have $784$ units corresponding to the $784$ pixels in each $28\times 28$ input image. A hidden layer $a^{[1]}$ will have $10$ units with ReLU activation, and finally our output layer $a^{[2]}$ will have $10$ units corresponding to the ten digit classes with softmax activation.

​ 我们的神经网络将有一个简单的两层结构。输入层 $a^{[0]}$ 将有 $784$ 个单元，对应于每个 $28\times 28$ 输入图像中的 $784$ 个像素。隐藏层 $a^{[1]}$ 将有 $10$ 个单元，用 ReLU 激活，最后我们的输出层 $a^{[2]}$ 将有 $10$ 个单元，对应于用 softmax 激活的 $10$ 个数字类别。

Vars and shapes

Forward prop

• $A^{[0]} = X$: 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

Backprop

• $dZ^{[2]}$: 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

---

初始化参数

使初始参数随机在 [-0.5, 0.5) 之间

通过 np.random.rand() 可以返回一个或一组服从“0 ~ 1”均匀分布的随机样本值。随机样本取值范围是 [0, 1)，不包括 1。

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

激活函数: ReLU

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

Softmax

• 一文详解Softmax函数

Softmax 函数将各个输出节点的输出值范围映射到 [0, 1]，并且约束各个输出节点的输出值的和为 1。

$Softmax(z_i)=\frac{e^z_i}{\sum^C_{C=1}e^{z_C}}$，其中 $z_i$ 为第 $i$ 个节点的输出值，$C$ 为输出结点的个数，即分类的类别个数。通过 Softmax 函数就可以将多分类的输出值转换为范围在 [0, 1] 和为 1 的概率分布。

def softmax(Z):
    A = np.exp(Z) / sum(np.exp(Z))
    return A

前向传播

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

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

ReLU 函数的导数，用于梯度下降

def ReLU_deriv(Z):
    return Z > 0

独热编码

将标签 Y 转为独热编码：

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

反向传播

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

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

调整参数

根据学习率 alpha 调整参数：

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

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

预测结果

numpy.argmax() 函数返回特定轴上数组的最大元素的索引，选取可能性最大的分类结果作为最终的分类结果。

def get_predictions(A2):
    return np.argmax(A2, 0)

计算准确率

def get_accuracy(predictions, Y):
    print(predictions, Y)
    return np.sum(predictions == Y) / Y.size  # 准确率 = 正确数 / 总数

梯度下降

def gradient_descent(X, Y, alpha, iterations):
    W1, b1, W2, b2 = init_params()  # 初始化参数
    for i in range(iterations):
        Z1, A1, Z2, A2 = forward_prop(W1, b1, W2, b2, X)  # 前向传播
        dW1, db1, dW2, db2 = backward_prop(Z1, A1, Z2, A2, W1, W2, X, Y)  # 反向传播
        W1, b1, W2, b2 = update_params(W1, b1, W2, b2, dW1, db1, dW2, db2, alpha)  # 调参
        if i % 10 == 0:  # 每训练 10 个 epoch 就显示一次准确率 
            print("Iteration:", i)
            predictions = get_predictions(A2)
            print("Accuray:", get_accuracy(predictions, Y))
    return W1, b1, W2, b2

训练神经网络

最后得到的准确率以及神经网络的各个参数:

W1, b1, W2, b2 = gradient_descent(X_train, Y_train, 0.10, 500)

Iteration: 0
[2 2 9 ... 9 2 2] [6 1 2 ... 5 3 1]
Accuray: 0.13534146341463416
Iteration: 10
[2 6 9 ... 3 6 2] [6 1 2 ... 5 3 1]
Accuray: 0.2577560975609756
Iteration: 20
[2 6 9 ... 3 1 2] [6 1 2 ... 5 3 1]
Accuray: 0.3676341463414634
Iteration: 30
[2 6 9 ... 3 1 1] [6 1 2 ... 5 3 1]
Accuray: 0.4432439024390244
Iteration: 40
[2 1 9 ... 3 1 1] [6 1 2 ... 5 3 1]
Accuray: 0.495390243902439
Iteration: 50
[2 1 8 ... 3 9 1] [6 1 2 ... 5 3 1]
Accuray: 0.5352682926829269
Iteration: 60
[2 1 8 ... 9 9 1] [6 1 2 ... 5 3 1]
Accuray: 0.568390243902439
Iteration: 70
[2 1 8 ... 9 9 1] [6 1 2 ... 5 3 1]
Accuray: 0.5975609756097561
Iteration: 80
[2 1 8 ... 9 9 1] [6 1 2 ... 5 3 1]
Accuray: 0.6207317073170732
Iteration: 90
[2 1 8 ... 9 9 1] [6 1 2 ... 5 3 1]
Accuray: 0.6396829268292683
Iteration: 100
[2 1 8 ... 5 9 1] [6 1 2 ... 5 3 1]
Accuray: 0.6570243902439025
Iteration: 110
[2 1 8 ... 5 9 1] [6 1 2 ... 5 3 1]
Accuray: 0.672829268292683
Iteration: 120
[2 1 8 ... 5 9 1] [6 1 2 ... 5 3 1]
Accuray: 0.6867560975609757
Iteration: 130
[2 1 8 ... 5 9 1] [6 1 2 ... 5 3 1]
Accuray: 0.6998048780487804
Iteration: 140
[2 1 8 ... 5 3 1] [6 1 2 ... 5 3 1]
Accuray: 0.710390243902439
Iteration: 150
[2 1 8 ... 5 3 1] [6 1 2 ... 5 3 1]
Accuray: 0.7207560975609756
Iteration: 160
[2 1 8 ... 5 3 1] [6 1 2 ... 5 3 1]
Accuray: 0.7307560975609756
Iteration: 170
[2 1 8 ... 5 3 1] [6 1 2 ... 5 3 1]
Accuray: 0.7396829268292683
Iteration: 180
[2 1 8 ... 5 3 1] [6 1 2 ... 5 3 1]
Accuray: 0.7484634146341463
Iteration: 190
[2 1 8 ... 5 3 1] [6 1 2 ... 5 3 1]
Accuray: 0.7559512195121951
Iteration: 200
[2 1 8 ... 5 3 1] [6 1 2 ... 5 3 1]
Accuray: 0.7630975609756098
Iteration: 210
[2 1 8 ... 5 3 1] [6 1 2 ... 5 3 1]
Accuray: 0.7700731707317073
Iteration: 220
[2 1 8 ... 5 3 1] [6 1 2 ... 5 3 1]
Accuray: 0.7764878048780488
Iteration: 230
[2 1 8 ... 5 3 1] [6 1 2 ... 5 3 1]
Accuray: 0.7819024390243903
Iteration: 240
[4 1 8 ... 5 3 1] [6 1 2 ... 5 3 1]
Accuray: 0.7870731707317074
Iteration: 250
[4 1 8 ... 5 3 1] [6 1 2 ... 5 3 1]
Accuray: 0.7922926829268293
Iteration: 260
[4 1 8 ... 5 3 1] [6 1 2 ... 5 3 1]
Accuray: 0.7969512195121952
Iteration: 270
[4 1 8 ... 5 3 1] [6 1 2 ... 5 3 1]
Accuray: 0.8013414634146342
Iteration: 280
[4 1 8 ... 5 3 1] [6 1 2 ... 5 3 1]
Accuray: 0.805439024390244
Iteration: 290
[4 1 8 ... 5 3 1] [6 1 2 ... 5 3 1]
Accuray: 0.8083414634146342
Iteration: 300
[4 1 8 ... 5 3 1] [6 1 2 ... 5 3 1]
Accuray: 0.8113658536585366
Iteration: 310
[4 1 8 ... 5 3 1] [6 1 2 ... 5 3 1]
Accuray: 0.8147560975609756
Iteration: 320
[4 1 8 ... 5 3 1] [6 1 2 ... 5 3 1]
Accuray: 0.818
Iteration: 330
[4 1 8 ... 5 3 1] [6 1 2 ... 5 3 1]
Accuray: 0.8207073170731707
Iteration: 340
[4 1 8 ... 5 3 1] [6 1 2 ... 5 3 1]
Accuray: 0.8234634146341463
Iteration: 350
[4 1 8 ... 5 3 1] [6 1 2 ... 5 3 1]
Accuray: 0.8260975609756097
Iteration: 360
[4 1 8 ... 5 3 1] [6 1 2 ... 5 3 1]
Accuray: 0.8283170731707317
Iteration: 370
[4 1 8 ... 5 3 1] [6 1 2 ... 5 3 1]
Accuray: 0.8308292682926829
Iteration: 380
[4 1 8 ... 5 3 1] [6 1 2 ... 5 3 1]
Accuray: 0.8332682926829268
Iteration: 390
[4 1 8 ... 5 3 1] [6 1 2 ... 5 3 1]
Accuray: 0.8349512195121951
Iteration: 400
[4 1 8 ... 5 3 1] [6 1 2 ... 5 3 1]
Accuray: 0.8371463414634146
Iteration: 410
[4 1 8 ... 5 3 1] [6 1 2 ... 5 3 1]
Accuray: 0.8391463414634146
Iteration: 420
[4 1 8 ... 5 3 1] [6 1 2 ... 5 3 1]
Accuray: 0.8407073170731707
Iteration: 430
[4 1 8 ... 5 3 1] [6 1 2 ... 5 3 1]
Accuray: 0.8424146341463414
Iteration: 440
[4 1 8 ... 5 3 1] [6 1 2 ... 5 3 1]
Accuray: 0.8438780487804878
Iteration: 450
[4 1 8 ... 5 3 1] [6 1 2 ... 5 3 1]
Accuray: 0.845390243902439
Iteration: 460
[4 1 8 ... 5 3 1] [6 1 2 ... 5 3 1]
Accuray: 0.8467317073170731
Iteration: 470
[4 1 8 ... 5 3 1] [6 1 2 ... 5 3 1]
Accuray: 0.8479756097560975
Iteration: 480
[4 1 8 ... 5 3 1] [6 1 2 ... 5 3 1]
Accuray: 0.848780487804878
Iteration: 490
[4 1 8 ... 5 3 1] [6 1 2 ... 5 3 1]
Accuray: 0.8501219512195122

最后得到 85.01% 的准确率。

训练结果可视化

将训练好的模型做出预测

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()  # matplotlib 库的 pyplot 模块中的 gray() 函数用于将颜色映射设置为“gray”。
    plt.imshow(current_image, interpolation='nearest')
    plt.show()

Let's look at a couple of examples:

test_prediction(0, W1, b1, W2, b2)
test_prediction(1, W1, b1, W2, b2)
test_prediction(2, W1, b1, W2, b2)
test_prediction(3, W1, b1, W2, b2)

Prediction:  [1]
Label:  1

Prediction:  [1]
Label:  1

Prediction:  [6]
Label:  6

Prediction:  [1]
Label:  1

将训练结果用于验证集

dev_predictions = make_predictions(X_dev, W1, b1, W2, b2)
get_accuracy(dev_predictions, Y_dev)

[6 3 3 4 3 5 9 2 2 0 6 4 9 5 9 4 1 2 7 9 0 0 3 1 1 2 5 1 0 6 8 4 6 4 1 1 0
 8 3 1 8 4 6 0 0 8 6 0 2 7 9 1 7 8 6 3 3 0 6 1 0 9 6 9 6 4 4 4 4 0 0 7 1 6
 6 0 6 4 7 9 6 1 6 1 5 5 0 2 9 9 3 9 4 9 7 7 9 9 1 1 1 6 0 8 3 7 8 6 0 2 5
 8 6 2 3 2 5 0 6 7 5 4 1 2 9 3 2 6 6 9 5 6 2 1 2 7 3 4 3 2 6 5 2 5 2 6 1 8
 1 7 4 8 4 1 2 2 0 1 8 1 5 2 6 8 5 7 8 1 0 0 0 9 2 7 5 6 5 6 9 7 4 9 4 6 7
 7 3 7 4 2 1 0 7 7 5 8 9 0 3 6 5 8 6 8 1 3 7 5 5 7 9 1 9 8 1 6 0 3 0 8 9 1
 1 7 9 4 1 9 3 3 1 6 0 2 8 2 4 6 8 1 0 9 6 3 3 2 6 1 4 0 8 2 7 0 2 2 1 2 7
 3 1 1 2 2 8 9 5 1 3 9 7 2 4 4 4 3 7 3 8 2 7 8 1 5 9 7 5 5 1 3 1 9 4 7 7 9
 4 1 4 7 9 2 9 3 1 2 7 7 0 3 3 8 7 5 6 7 9 4 7 3 3 9 2 3 2 8 9 2 3 6 0 5 4
 3 2 7 0 4 2 5 4 8 9 9 2 2 7 8 4 1 6 3 2 9 6 2 4 3 7 3 6 6 4 5 1 2 0 9 1 5
 5 6 9 5 8 5 6 8 9 0 9 8 6 0 7 8 0 5 0 3 2 3 3 9 4 1 4 6 6 9 6 1 3 3 3 0 6
 5 8 0 1 6 5 6 1 8 9 8 0 2 1 3 3 9 9 6 2 9 8 2 3 6 5 6 9 7 0 7 4 8 3 4 9 9
 4 3 2 0 2 4 5 8 9 9 5 0 7 3 0 4 9 1 2 1 7 7 6 2 2 0 6 1 9 8 8 0 0 1 9 3 9
 8 4 8 8 1 4 2 0 1 3 5 1 8 8 3 9 7 1 1 2 6 2 6 1 7 1 2 5 8 0 3 0 9 5 8 9 0
 0 8 1 9 3 1 4 4 4 9 7 2 8 1 5 5 8 5 4 4 1 8 5 6 1 1 9 2 8 3 7 5 7 4 9 4 3
 6 7 9 0 8 0 1 7 2 8 7 1 3 8 4 4 1 1 9 7 1 8 4 4 0 3 3 2 6 8 8 7 7 7 5 3 0
 4 5 5 7 2 3 0 1 7 1 2 8 0 8 0 6 6 6 6 6 4 5 5 8 0 1 3 5 7 7 4 8 6 9 1 2 4
 4 4 6 5 0 2 3 1 3 8 1 4 4 7 9 9 9 6 0 5 9 9 6 9 6 4 3 1 1 0 0 5 9 6 4 1 0
 1 7 9 9 1 1 1 4 4 3 8 2 1 0 8 5 0 9 0 2 8 5 2 0 3 7 6 3 0 4 3 9 2 5 2 3 2
 4 4 6 7 1 6 7 1 7 0 3 7 3 6 2 8 2 1 6 4 1 5 8 6 8 7 5 1 6 8 3 1 8 2 9 1 8
 6 7 0 4 2 6 8 9 7 2 8 6 4 2 6 3 8 5 1 8 3 3 0 1 4 1 0 9 1 7 3 6 0 4 2 1 7
 1 0 0 0 2 0 9 7 9 8 4 3 6 6 9 0 5 8 5 0 6 3 1 9 2 5 2 7 4 8 6 7 7 9 3 9 4
 3 6 3 5 2 1 4 5 5 9 6 5 1 8 5 2 2 2 0 5 8 6 2 7 7 5 2 4 0 2 6 4 2 4 7 9 8
 3 6 0 2 8 9 4 6 6 8 7 2 2 7 2 0 2 9 5 2 1 3 7 6 2 0 7 4 7 6 0 6 6 0 1 5 1
 2 9 3 9 8 2 9 6 5 4 9 3 7 8 1 8 4 7 7 0 2 3 5 5 7 5 3 4 9 0 8 2 4 3 0 9 7
 0 2 3 7 5 2 0 2 7 5 9 6 8 9 1 2 7 6 0 1 4 6 4 3 8 4 2 6 1 0 4 7 3 0 7 2 5
 5 1 8 6 4 6 1 0 6 6 1 8 1 3 9 9 8 0 4 4 4 7 0 1 0 9 0 5 4 1 5 4 4 7 0 4 7
 1] [6 3 3 4 3 5 9 2 9 0 6 4 9 5 9 4 1 2 7 9 0 0 3 1 1 2 5 1 2 6 8 2 6 4 1 1 0
 8 3 1 2 4 6 0 0 8 6 0 2 7 9 1 7 8 6 3 3 0 6 1 0 9 0 9 6 4 4 4 4 0 0 7 1 6
 6 0 6 4 7 9 6 1 6 1 5 5 0 2 9 9 3 9 4 4 7 7 9 9 1 1 1 6 0 8 3 7 8 6 0 2 0
 8 6 4 3 3 5 0 6 7 5 4 1 2 4 3 2 6 6 9 5 6 2 1 2 7 3 8 9 2 6 5 2 5 2 6 1 9
 1 7 4 8 4 1 5 2 0 1 8 1 3 2 5 8 5 7 5 1 0 0 0 9 2 7 5 6 9 6 7 7 4 9 4 6 7
 7 3 7 4 2 1 0 7 7 5 5 9 0 3 6 5 3 6 8 1 3 9 5 5 7 4 1 4 5 8 6 0 3 0 5 9 1
 7 9 3 4 7 9 3 3 1 6 0 2 5 2 4 0 8 1 2 9 6 3 8 2 6 1 9 0 8 3 7 0 8 2 1 2 7
 3 1 1 2 2 8 7 5 1 3 9 7 3 4 4 4 3 7 8 8 7 7 3 1 5 9 7 5 5 1 3 1 9 4 7 2 9
 4 1 4 7 9 2 9 3 1 2 7 7 0 9 8 8 7 5 6 7 5 4 7 3 3 9 2 3 2 8 9 2 3 6 0 5 4
 3 2 7 0 4 2 5 4 8 9 9 2 2 7 4 4 1 6 3 6 4 6 2 4 8 7 3 6 6 2 5 1 2 0 4 3 3
 5 6 9 5 8 5 6 8 9 8 9 8 6 5 7 9 0 5 0 8 2 3 9 4 4 1 4 6 6 9 6 1 3 3 3 0 6
 5 3 0 1 4 5 6 1 7 9 8 6 2 1 3 3 9 9 8 7 9 8 4 3 6 5 6 9 7 0 3 9 8 3 4 9 9
 4 3 2 0 3 4 5 8 9 9 0 0 7 5 0 4 8 1 2 1 7 7 6 2 2 0 6 1 9 5 5 0 4 8 4 5 9
 2 4 8 8 5 4 2 0 1 1 5 1 8 8 3 9 7 1 1 8 6 2 6 1 9 1 3 5 8 6 3 0 9 5 2 9 0
 0 8 1 9 3 1 4 4 6 9 7 2 8 1 5 5 8 5 4 4 2 1 5 6 1 1 9 2 8 3 7 5 6 4 9 4 3
 6 7 9 0 8 0 1 7 2 8 9 1 3 8 4 4 1 1 9 7 1 8 4 4 0 3 3 2 6 8 8 7 7 7 5 3 0
 4 7 5 7 2 3 0 1 7 1 2 8 0 8 0 6 2 6 5 6 4 5 5 8 0 1 5 5 7 7 4 8 6 9 1 2 4
 4 4 6 5 0 6 5 1 3 8 1 4 4 7 8 9 9 6 0 5 9 9 6 9 8 4 9 1 1 0 0 5 9 6 4 1 7
 1 7 7 9 1 1 1 4 5 3 8 4 1 0 8 5 0 9 0 6 8 8 2 0 3 7 6 3 0 4 3 9 3 5 7 3 3
 4 4 6 7 1 6 7 1 7 0 3 7 3 6 1 2 2 1 6 4 1 5 8 5 8 7 5 5 6 8 3 1 8 6 8 1 8
 6 7 0 4 2 6 8 7 7 2 8 6 4 3 6 3 8 5 1 8 3 3 0 1 4 1 0 7 1 7 3 6 0 4 3 1 7
 1 0 0 0 2 0 9 7 4 8 4 3 8 6 4 0 5 8 5 0 6 3 1 9 2 5 9 7 4 5 6 7 2 9 3 9 4
 3 6 6 5 9 1 4 5 5 9 2 5 1 8 5 2 2 2 0 3 5 6 2 7 7 5 7 9 0 2 6 8 8 4 7 9 8
 5 6 0 6 8 7 4 8 6 9 7 2 2 7 2 0 2 9 5 1 1 3 4 6 2 0 7 4 7 6 0 6 6 0 1 5 1
 2 9 3 9 9 2 9 6 5 4 4 7 7 8 1 8 4 7 7 0 8 3 5 5 7 3 3 4 9 0 8 2 4 3 0 9 7
 0 2 3 7 5 2 0 2 7 5 9 6 8 9 1 2 8 6 0 1 4 2 4 3 8 4 2 6 1 0 4 7 3 0 3 2 5
 0 1 8 6 4 6 1 0 6 6 1 8 1 5 9 9 8 2 4 4 4 7 0 1 0 9 0 5 4 1 5 4 4 7 0 4 7
 8]



0.845

Still 84% accuracy, so our model generalized from the training data pretty well.

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