我试图在 python 中从头开始实现逻辑回归，以更好地了解它是如何工作的。特别是我正在关注Andrew Ng 的这个视频教程。

这是我用于测试算法的数据集：marks.txt

我发现如果不对数据进行归一化，算法不会收敛，损失也不会减少（有时是 NaN）。

这是实现：

import numpy as np
import pandas as pd


def normalize(X, axis=0):
    return (X - np.mean(X, axis=axis)) / np.std(X, axis=axis)


class LogisticRegression():
    def __init__(self, num_epochs=10000, lr=0.001 ):
        self.num_epochs = num_epochs
        self.lr = lr

    def __sigmoid(self, Z):
        return 1 / (1 + np.exp(-Z))

    def __loss(self, y, A):
        return - np.mean(y * np.log(A) + (1 - y) * np.log(1 - A))


    def fit(self, X, y):
        n, m = X.shape
        print(f'Number of features = {n}')
        print(f'Number of samples = {m}')

        W = np.zeros((n, 1))
        b = 0

        for epoch in range(1, self.num_epochs + 1):
            Z = np.dot(W.T, X) + b
            A = self.__sigmoid(Z)

            dZ = A - y
            dW = 1/m * np.dot(X, dZ.T)
            db = 1/m * np.sum(dZ)

            W -= self.lr * dW
            b -= self.lr * db

            if epoch == 1 or epoch % 100 == 0:
                J = self.__loss(y, A)
                print(f'Epoch {epoch} - Loss = {J}')


columns = [
    'mark_1',
    'mark_2',
    'y'
]

data = pd.read_csv('marks.txt', names=columns, header=None)

X = data.iloc[:, :-1].values
y = data.iloc[:, -1:].values

lr = LogisticRegression(num_epochs=10000, lr=0.01)
lr.fit(X.T, y.T)

如果我执行这个，我得到以下输出：

Number of features = 2
Number of samples = 100
Epoch 1 - Loss = 0.6931471805599453
Epoch 100 - Loss = nan
Epoch 200 - Loss = 4.804976603222295
Epoch 300 - Loss = 7.859811065112183
Epoch 400 - Loss = nan
Epoch 500 - Loss = 4.7897185742553186
Epoch 600 - Loss = 7.836867515204696
Epoch 700 - Loss = nan
Epoch 800 - Loss = 4.774454897975551
Epoch 900 - Loss = 7.813880674612202
Epoch 1000 - Loss = nan
Epoch 1100 - Loss = 4.759205019552172
Epoch 1200 - Loss = 7.790844866695895
Epoch 1300 - Loss = nan
Epoch 1400 - Loss = 4.743971469023722
Epoch 1500 - Loss = 7.7677542901506875
Epoch 1600 - Loss = nan
Epoch 1700 - Loss = 4.728752106484584
Epoch 1800 - Loss = 7.744603001274255
Epoch 1900 - Loss = nan
Epoch 2000 - Loss = 4.713554131813897
Epoch 2100 - Loss = 7.721384895223854
Epoch 2200 - Loss = nan
Epoch 2300 - Loss = 4.698361852011675
Epoch 2400 - Loss = 7.698093686352158
Epoch 2500 - Loss = nan
Epoch 2600 - Loss = 4.683196015386
Epoch 2700 - Loss = 7.674722887733128
Epoch 2800 - Loss = nan
Epoch 2900 - Loss = 4.6680544815204925
Epoch 3000 - Loss = 7.6512657900116805
Epoch 3100 - Loss = nan
Epoch 3200 - Loss = 4.65294492110793
Epoch 3300 - Loss = 7.627715439737382
Epoch 3400 - Loss = nan
Epoch 3500 - Loss = 4.637870088341966
Epoch 3600 - Loss = 7.604064617373076
Epoch 3700 - Loss = nan
Epoch 3800 - Loss = 4.62281379407897
Epoch 3900 - Loss = 7.580305815203287
Epoch 4000 - Loss = nan
Epoch 4100 - Loss = 4.60781250038029
Epoch 4200 - Loss = 7.556431215405509
Epoch 4300 - Loss = nan
Epoch 4400 - Loss = 4.592835472351133
Epoch 4500 - Loss = 7.532432668589291__main__:19: RuntimeWarning: divide by zero encountered in log
__main__:19: RuntimeWarning: invalid value encountered in multiply

Epoch 4600 - Loss = nan
Epoch 4700 - Loss = 4.57789045326783
Epoch 4800 - Loss = 7.508301673152992
Epoch 4900 - Loss = nan
Epoch 5000 - Loss = 4.563010160089178
Epoch 5100 - Loss = 7.48402935585585
Epoch 5200 - Loss = nan
Epoch 5300 - Loss = 4.548178514140011
Epoch 5400 - Loss = 7.459606454052721
Epoch 5500 - Loss = nan
Epoch 5600 - Loss = 4.533383810118562
Epoch 5700 - Loss = 7.4350233000888215
Epoch 5800 - Loss = nan
Epoch 5900 - Loss = 4.518654642394596
Epoch 6000 - Loss = 7.4102698084014715
Epoch 6100 - Loss = nan
Epoch 6200 - Loss = 4.503979863783454
Epoch 6300 - Loss = 7.385335465922019
Epoch 6400 - Loss = nan
Epoch 6500 - Loss = 4.489373494021938
Epoch 6600 - Loss = 7.3602093264129
Epoch 6700 - Loss = nan
Epoch 6800 - Loss = 4.474823890959029
Epoch 6900 - Loss = 7.334880009407734
Epoch 7000 - Loss = nan
Epoch 7100 - Loss = 4.46034588627989
Epoch 7200 - Loss = 7.309335704445365
Epoch 7300 - Loss = nan
Epoch 7400 - Loss = 4.445943673235942
Epoch 7500 - Loss = 7.283564181295987
Epoch 7600 - Loss = nan
Epoch 7700 - Loss = 4.431627025808597
Epoch 7800 - Loss = 7.257552806867497
Epoch 7900 - Loss = nan
Epoch 8000 - Loss = 4.4174006755844095
Epoch 8100 - Loss = 7.231288569447483
Epoch 8200 - Loss = nan
Epoch 8300 - Loss = 4.403243132948716
Epoch 8400 - Loss = 7.2047581108777345
Epoch 8500 - Loss = nan
Epoch 8600 - Loss = 4.3891854547270475
Epoch 8700 - Loss = 7.177947767170446
Epoch 8800 - Loss = nan
Epoch 8900 - Loss = 4.3752278129468944
Epoch 9000 - Loss = 7.150843617955713
Epoch 9100 - Loss = nan
Epoch 9200 - Loss = 4.361364188554799
Epoch 9300 - Loss = 7.123431544995584
Epoch 9400 - Loss = nan
Epoch 9500 - Loss = 4.34760573682999
Epoch 9600 - Loss = 7.095697299812474
Epoch 9700 - Loss = nan
Epoch 9800 - Loss = 4.333968348729086
Epoch 9900 - Loss = 7.067626580257764
Epoch 10000 - Loss = nan

否则，如果我在拟合模型之前对数据进行归一化（零均值单位方差），它似乎可以正常工作（我可以看到损失减少）：

def normalize(X, axis=0):
    return (X - np.mean(X, axis=axis)) / np.std(X, axis=axis)

# [...]

data = pd.read_csv('data/marks.csv', names=columns, header=0)

X = data.iloc[:, :-1].values
y = data.iloc[:, -1:].values

X = normalize(X)  # Normalize data  

lr = LogisticRegression(num_epochs=10000, lr=0.01)
lr.fit(X.T, y.T)

输出：

Number of features = 2
Number of samples = 100
Epoch 1 - Loss = 0.6931471805599453
Epoch 100 - Loss = 0.5733935847364559
Epoch 200 - Loss = 0.4967653811151946
Epoch 300 - Loss = 0.4456019909522728
Epoch 400 - Loss = 0.4094643825544129
Epoch 500 - Loss = 0.38268993233906584
Epoch 600 - Loss = 0.3620711093033572
Epoch 700 - Loss = 0.34569287144258726
Epoch 800 - Loss = 0.33235335562007345
Epoch 900 - Loss = 0.3212643235482712
Epoch 1000 - Loss = 0.3118887927384652
Epoch 1100 - Loss = 0.3038488197031603
Epoch 1200 - Loss = 0.29687082702689194
Epoch 1300 - Loss = 0.29075191833313574
Epoch 1400 - Loss = 0.28533840424281665
Epoch 1500 - Loss = 0.28051169317796093
Epoch 1600 - Loss = 0.2761787696062344
Epoch 1700 - Loss = 0.27226561328471377
Epoch 1800 - Loss = 0.2687125532958896
Epoch 1900 - Loss = 0.2654709248024843
Epoch 2000 - Loss = 0.2625006214821513
Epoch 2100 - Loss = 0.25976827555417215
Epoch 2200 - Loss = 0.25724588518032776
Epoch 2300 - Loss = 0.25490976581258473
Epoch 2400 - Loss = 0.2527397395020597
Epoch 2500 - Loss = 0.2507185013239025
Epoch 2600 - Loss = 0.24883111923883924
Epoch 2700 - Loss = 0.24706463561630515
Epoch 2800 - Loss = 0.24540774701833637
Epoch 2900 - Loss = 0.24385054481312596
Epoch 3000 - Loss = 0.24238430349560172
Epoch 3100 - Loss = 0.24100130673780307
Epoch 3200 - Loss = 0.23969470351293196
Epoch 3300 - Loss = 0.2384583883670421
Epoch 3400 - Loss = 0.23728690121408835
Epoch 3500 - Loss = 0.23617534301823132
Epoch 3600 - Loss = 0.23511930448367577
Epoch 3700 - Loss = 0.23411480545586144
Epoch 3800 - Loss = 0.23315824319134465
Epoch 3900 - Loss = 0.2322463480086729
Epoch 4000 - Loss = 0.23137614511221863
Epoch 4100 - Loss = 0.23054492160266965
Epoch 4200 - Loss = 0.2297501978647279
Epoch 4300 - Loss = 0.22898970266443042
Epoch 4400 - Loss = 0.22826135140291767
Epoch 4500 - Loss = 0.22756322706622129
Epoch 4600 - Loss = 0.22689356348620354
Epoch 4700 - Loss = 0.22625073058962492
Epoch 4800 - Loss = 0.22563322136316274
Epoch 4900 - Loss = 0.22503964030418896
Epoch 5000 - Loss = 0.22446869316192905
Epoch 5100 - Loss = 0.22391917780259743
Epoch 5200 - Loss = 0.22338997605632013
Epoch 5300 - Loss = 0.2228800464239599
Epoch 5400 - Loss = 0.22238841753904443
Epoch 5500 - Loss = 0.2219141822944289
Epoch 5600 - Loss = 0.22145649255554345
Epoch 5700 - Loss = 0.22101455439246653
Epoch 5800 - Loss = 0.22058762377191163
Epoch 5900 - Loss = 0.22017500265778595
Epoch 6000 - Loss = 0.21977603547546334
Epoch 6100 - Loss = 0.21939010590048816
Epoch 6200 - Loss = 0.21901663393723186
Epoch 6300 - Loss = 0.21865507325717176
Epoch 6400 - Loss = 0.2183049087700573
Epoch 6500 - Loss = 0.21796565440434787
Epoch 6600 - Loss = 0.21763685107601996
Epoch 6700 - Loss = 0.2173180648272089
Epoch 6800 - Loss = 0.2170088851182173
Epoch 6900 - Loss = 0.2167089232582348
Epoch 7000 - Loss = 0.2164178109617001
Epoch 7100 - Loss = 0.21613519901863631
Epoch 7200 - Loss = 0.21586075606851618
Epoch 7300 - Loss = 0.21559416746830468
Epoch 7400 - Loss = 0.21533513424628176
Epoch 7500 - Loss = 0.2150833721340996
Epoch 7600 - Loss = 0.21483861067028356
Epoch 7700 - Loss = 0.2146005923690525
Epoch 7800 - Loss = 0.21436907194893295
Epoch 7900 - Loss = 0.21414381561617263
Epoch 8000 - Loss = 0.21392460039843075
Epoch 8100 - Loss = 0.21371121352464903
Epoch 8200 - Loss = 0.21350345184738453
Epoch 8300 - Loss = 0.213301121304228
Epoch 8400 - Loss = 0.21310403641523415
Epoch 8500 - Loss = 0.21291201981356775
Epoch 8600 - Loss = 0.21272490180681325
Epoch 8700 - Loss = 0.2125425199666215
Epoch 8800 - Loss = 0.21236471874456622
Epoch 8900 - Loss = 0.21219134911226462
Epoch 9000 - Loss = 0.21202226822398146
Epoch 9100 - Loss = 0.21185733910008378
Epoch 9200 - Loss = 0.21169643032984978
Epoch 9300 - Loss = 0.21153941579225452
Epoch 9400 - Loss = 0.21138617439347157
Epoch 9500 - Loss = 0.211236589819924
Epoch 9600 - Loss = 0.21109055030581808
Epoch 9700 - Loss = 0.21094794841416828
Epoch 9800 - Loss = 0.2108086808304075
Epoch 9900 - Loss = 0.21067264816773867
Epoch 10000 - Loss = 0.21053975478345366

我可以看到两个可能的原因：

• 我的实现是错误的。
• 规范化对于使算法正常工作是强制性的（但我​​在这里读到它只是推荐的，不是强制性的）。

如果在实现中存在问题或任何其他考虑为什么算法在没有规范化的情况下无法工作，任何人都可以给出提示吗？