Demystifying Neural Networks (Part 2): Gradient Descent and Cost Functions, Making Your Model Smarter | 揭秘神经网络（二）：梯度下降与成本函数，让你的模型更聪明

在上一篇博客中，我们了解了神经网络的基本结构和工作原理。

In the previous blog post, we learned about the basic structure and working principles of neural networks.

今天，我们将深入探讨神经网络如何学习，以及两个关键概念：成本函数和梯度下降。

Today, we will delve into how neural networks learn and explore two key concepts: cost functions and gradient descent.

1. 神经网络如何学习？

1. How do Neural Networks Learn?

想象一下，你是一个新手司机，正在学习如何驾驶汽车。一开始，你可能会犯很多错误，比如转弯太急、刹车太晚等等。但是，通过不断练习和从错误中学习，你会逐渐掌握驾驶技巧，变得越来越熟练。

Imagine you are a novice driver learning how to drive a car. At first, you might make many mistakes, such as turning too sharply or braking too late. However, through continuous practice and learning from your mistakes, you will gradually master driving skills and become more proficient.

神经网络的学习过程与之类似。它们通过不断调整内部参数（权重和偏置），来减少预测误差，从而提高预测的准确性。这个过程就叫做训练。

The learning process of neural networks is similar. They continuously adjust their internal parameters (weights and biases) to reduce prediction errors, thus improving prediction accuracy. This process is called training.

2. 成本函数：衡量预测误差的标尺

2. Cost Function: A Measure of Prediction Error

在训练神经网络时，我们需要一个指标来衡量模型的预测结果与真实值之间的差距，这个指标就是成本函数。成本函数的值越小，说明模型的预测越准确。

When training a neural network, we need a metric to measure the difference between the predicted results of the model and the true values. This metric is called the cost function. The smaller the value of the cost function, the more accurate the model's prediction.

成本函数的参数与返回值： Parameters and Return Values of the Cost Function:

• 参数： 成本函数的输入参数通常包括模型的预测值（y_hat）和真实值（y）。
• Parameters: The input parameters of the cost function usually include the predicted value (y_hat) of the model and the true value (y).
• 返回值： 成本函数的输出值是一个标量（单个数字），表示模型预测值与真实值之间的差异程度。输出值越大，说明模型的预测越不准确；输出值越小，说明模型的预测越准确。
• Return Value: The output value of the cost function is a scalar (single number) that represents the degree of difference between the predicted value and the true value of the model. The larger the output value, the less accurate the model's prediction; the smaller the output value, the more accurate the model's prediction.

常见的成本函数： Common Cost Functions:

• 均方误差 (Mean Squared Error, MSE): 适用于回归问题。它计算预测值与真实值之间差的平方的平均值。
• Mean Squared Error (MSE): Used for regression problems. It calculates the average of the squares of the differences between the predicted and true values.

MSE = (1/n) * Σ(y_hat - y)^2

• 交叉熵 (Cross Entropy): 适用于分类问题。它衡量预测的概率分布与真实概率分布之间的差异。
• Cross Entropy: Used for classification problems. It measures the difference between the predicted probability distribution and the true probability distribution.

Cross Entropy = -Σ y * log(y_hat)

成本函数的作用： The Role of the Cost Function:

成本函数就像一个导航仪，它告诉我们模型当前的预测有多好，以及应该向哪个方向调整参数才能让预测更准确。 The cost function is like a navigation device, it tells us how good the model's current prediction is and in which direction the parameters should be adjusted to make the prediction more accurate.

3. 梯度下降：寻找最优参数的“导航员”

3. Gradient Descent: The "Navigator" for Finding Optimal Parameters

梯度下降是一种优化算法，它可以帮助我们找到成本函数的最小值，从而找到最优的模型参数（权重和偏置）。 Gradient descent is an optimization algorithm that helps us find the minimum value of the cost function, thereby finding the optimal model parameters (weights and biases).

梯度下降的原理： The Principle of Gradient Descent:

1. 计算梯度： 梯度是成本函数对每个参数的偏导数，它表示了改变参数对成本函数的影响程度。
   1. Calculate Gradient: The gradient is the partial derivative of the cost function with respect to each parameter. It represents the degree of influence of changing the parameter on the cost function.
2. 更新参数： 沿着梯度的反方向更新参数，即朝着成本函数减小的方向调整参数。 2. Update Parameters: Update the parameters in the opposite direction of the gradient, i.e., adjust the parameters in the direction of decreasing the cost function.
3. 重复迭代： 重复上述步骤，直到成本函数的值收敛到最小值。 3. Repeat Iteration: Repeat the above steps until the value of the cost function converges to the minimum.

形象比喻： Analogy:

想象你站在一座山上，想要下山。梯度下降就是帮助你找到最陡峭的下山路径，然后沿着这条路径一步步往下走，最终到达山谷底部（成本函数的最小值）。

Imagine you are standing on a mountain and want to go down. Gradient descent is like helping you find the steepest path down the mountain, and then walking down this path step by step until you reach the bottom of the valley (the minimum value of the cost function).

数学公式： Mathematical Formula:

θ_new = θ_old - α * ∇J(θ_old)

其中： / Where:

• θ_new 是更新后的参数（权重或偏置）
• θ_new is the updated parameter (weight or bias)
• θ_old 是更新前的参数
• θ_old is the parameter before updating
• α 是学习率，控制每次更新的步长
• α is the learning rate, which controls the step size of each update
• ∇J(θ_old) 是成本函数在 θ_old 处的梯度
• ∇J(θ_old) is the gradient of the cost function at θ_old

学习率的选择： Choice of Learning Rate:

学习率是一个重要的超参数，它控制着每次参数更新的步长。学习率过大可能导致模型不稳定，学习率过小可能导致模型收敛速度慢。

The learning rate is an important hyperparameter that controls the step size of each parameter update. A learning rate that is too large may lead to an unstable model, while a learning rate that is too small may lead to slow convergence of the model.

4. 总结

4. Summary

成本函数和梯度下降是神经网络训练的两个核心概念。成本函数告诉我们模型的预测有多好，梯度下降告诉我们如何改进模型。通过不断迭代，神经网络可以逐渐学习到数据中的模式，提高预测的准确性。

Cost function and gradient descent are two core concepts in neural network training. The cost function tells us how good the model's prediction is, and gradient descent tells us how to improve the model. Through continuous iteration, neural networks can gradually learn patterns in the data and improve prediction accuracy.

希望这篇博客能帮助你更清晰地理解成本函数和梯度下降。如果你还有其他问题，请随时提出。

I hope this blog post can help you understand cost function and gradient descent more clearly. If you have any other questions, please feel free to ask.