Theory and techniques of weight update in neural networks

Weight update in neural network is to adjust the connection weights between neurons in the network through methods such as back propagation algorithm to improve the performance of the network. This article will introduce the concept and method of weight update to help readers better understand the training process of neural networks.

1. Concept

The weights in neural networks are parameters connecting different neurons and determine the strength of signal transmission. Each neuron receives the signal from the previous layer, multiplies it by the weight of the connection, adds a bias term, and is finally activated through the activation function and passed to the next layer. Therefore, the size of the weight directly affects the strength and direction of the signal, which in turn affects the output of the neural network.

The purpose of weight update is to optimize the performance of the neural network. During the training process, the neural network adapts to the training data by continuously adjusting the weights between neurons to improve the prediction ability on the test data. By adjusting the weights, the neural network can better fit the training data, thereby improving the prediction accuracy. In this way, the neural network can more accurately predict the results of unknown data and achieve better performance.

2. Method

Commonly used weight update methods in neural networks include gradient descent, stochastic gradient descent, and batch gradient descent.

Gradient descent method

The gradient descent method is one of the most basic weight update methods. Its basic idea is to calculate the loss function to update the weight. The gradient (that is, the derivative of the loss function with respect to the weight) is used to update the weight to minimize the loss function. Specifically, the steps of the gradient descent method are as follows:

First, we need to define a loss function to measure the performance of the neural network on the training data. Usually, we will choose the mean square error (MSE) as the loss function, which is defined as follows:

MSE=\frac{1}{n}\sum_{i=1} ^{n}(y_i-\hat{y_i})^2

Where, y_i represents the true value of the i-th sample, \hat{y_i} represents the neural network's response to the i-th sample The predicted value of samples, n represents the total number of samples.

Then, we need to calculate the derivative of the loss function with respect to the weight, that is, the gradient. Specifically, for each weight w_{ij} in the neural network, its gradient can be calculated by the following formula:

\frac{\partial MSE}{\partial w_{ij }}=\frac{2}{n}\sum_{k=1}^{n}(y_k-\hat{y_k})\cdot f'(\sum_{j=1}^{m}w_{ij }x_{kj})\cdot x_{ki}

Among them, n represents the total number of samples, m represents the input layer size of the neural network, and x_{kj} represents the kth sample For the jth input feature, f(\cdot) represents the activation function, and f'(\cdot) represents the derivative of the activation function.

Finally, we can update the weights through the following formula:

w_{ij}=w_{ij}-\alpha\cdot\ frac{\partial MSE}{\partial w_{ij}}

Among them, \alpha represents the learning rate, which controls the step size of weight update.

Stochastic gradient descent method

The stochastic gradient descent method is a variant of the gradient descent method. Its basic idea is to randomly select each time A sample is used to calculate the gradient and update the weights. Compared to the gradient descent method, the stochastic gradient descent method can converge faster and be more efficient when processing large-scale data sets. Specifically, the steps of the stochastic gradient descent method are as follows:

First, we need to shuffle the training data and randomly select a sample x_k to calculate the gradient. We can then calculate the derivative of the loss function with respect to the weights via the following formula:

\frac{\partial MSE}{\partial w_{ij}}=2(y_k-\hat {y_k})\cdot f'(\sum_{j=1}^{m}w_{ij}x_{kj})\cdot x_{ki}

where, y_k represents the true value of the k-th sample, \hat{y_k} represents the predicted value of the k-th sample by the neural network.

Finally, we can update the weights through the following formula:

w_{ij}=w_{ij}-\alpha\cdot\ frac{\partial MSE}{\partial w_{ij}}

Among them, \alpha represents the learning rate, which controls the step size of weight update.

Batch gradient descent method

The batch gradient descent method is another variant of the gradient descent method. The basic idea is to use A mini-batch of samples is used to calculate the gradient and update the weights. Compared with gradient descent and stochastic gradient descent, batch gradient descent can converge more stably and is more efficient when processing small-scale data sets. Specifically, the steps of the batch gradient descent method are as follows:

First, we need to divide the training data into several mini-batches of equal size, each mini-batch contains b samples. We can then calculate the average gradient of the loss function against the weights on each mini-batch, which is:

\frac{1}{b}\sum_{k=1}^{ b}\frac{\partial MSE}{\partial w_{ij}}

where b represents the mini-batch size. Finally, we can update the weights through the following formula:

w_{ij}=w_{ij}-\alpha\cdot\frac{1}{b}\sum_{k= 1}^{b}\frac{\partial MSE}{\partial w_{ij}}

Among them, \alpha represents the learning rate, which controls the step size of weight update.

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