The impact of feature scaling on local optimal solutions

Feature scaling plays an important role in machine learning and has a close relationship with local optimality. Feature scaling refers to scaling feature data so that they have a similar numerical range. The purpose of this is to avoid certain features from having an excessive impact on the results during model training, thereby making the model more stable and accurate. Local optimality refers to the optimal solution found in a local area, but it is not necessarily the global optimal solution. In machine learning, optimization algorithms often find the optimal solution iteratively. If the range of feature data differs greatly, then during the model training process, some features may have a greater impact on the convergence of the optimization algorithm, causing the algorithm to fall into local optimality and fail to find the global optimal solution. To solve this problem, we can scale the features. By scaling the feature data to similar

The purpose of feature scaling is to ensure that the numerical ranges of different features are similar and to avoid certain features from having an excessive impact on the model training results.

Suppose we have a simple linear regression problem, characterized by house area (unit: square meters) and house price (unit: 10,000 yuan). If we do not scale the features and directly use the original data for modeling, we may encounter local optimal problems. This is because the numerical ranges of features may be different, causing the model to favor features with larger values ​​in calculations. In order to solve this problem, we can scale the features, such as using mean normalization or standardization, to scale the feature values ​​to the same value range. This ensures that the model gives the same importance to all features when calculating,

import numpy as np
from sklearn.linear_model import LinearRegression

# 原始数据
area = np.array([100, 150, 200, 250, 300]).reshape(-1, 1)
price = np.array([50, 75, 100, 125, 150])

# 不进行特征缩放的线性回归
model_unscaled = LinearRegression()
model_unscaled.fit(area, price)

# 缩放数据
area_scaled = (area - np.mean(area)) / np.std(area)
price_scaled = (price - np.mean(price)) / np.std(price)

# 进行特征缩放的线性回归
model_scaled = LinearRegression()
model_scaled.fit(area_scaled, price_scaled)

In the above code, we first use the non-feature-scaled data for linear regression modeling, and then use the feature-scaled data Perform linear regression modeling on the data.

Since the units of area and price are different, the linear regression algorithm may fit the area feature more significantly and ignore the price. Feature scaling is necessary to avoid a poor fit of the model near the local optimum.

This problem can be avoided by scaling the features so that the two features have the same scale. By performing linear regression modeling on the feature-scaled data, the model can treat the two features more evenly, reducing the problem of local optimal points caused by the influence of different scales.

It should be noted that the feature scaling in the code uses mean normalization and standardization, and the appropriate feature scaling method can be selected according to the actual situation.

In summary, feature scaling helps avoid local optimality. By unifying the scale, it ensures the balance of feature weights and improves the model's ability to better get rid of local optimal points during the training process. , thereby improving the possibility of overall optimization.

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