introduce
Gradient descent is a commonly used optimization algorithm for finding the minimum or maximum of a function. In the field of machine learning and deep learning, gradient descent is widely used in the training process of the model to optimize the model parameters by minimizing the loss function, so that the model can better fit the training data.
Fundamental
The basic principle of the gradient descent algorithm is to continuously adjust the value of the parameters in an iterative manner, so that the value of the objective function gradually approaches the optimal solution. In optimization problems, we want to find the minimum value of the objective function, so the gradient descent algorithm will update the parameters in the opposite direction of the gradient (or derivative) to reduce the value of the objective function. Specifically, for a parameter vector θ θθ and the objective functionJ ( θ ) J(θ)J ( θ ) , the update formula of gradient descent is:
θ = θ − α ∗ ∇ J ( θ ) θ = θ - α * ∇J(θ)i=i−a∗∇ J ( θ )
Among them, α αα is the learning rate (learning rate), which controls the step size of the gradient descent. Too large a learning rate may lead to oscillation and instability, while too small a learning rate may lead to slow convergence. Therefore, choosing an appropriate learning rate is an important tuning parameter for the gradient descent algorithm.
batch gradient descent
Batch gradient descent is the most basic form of gradient descent algorithm, which uses all training samples to calculate gradients in each iteration and then update parameters. Due to the need to traverse the entire training set, batch gradient descent is computationally expensive, especially on large-scale datasets. But its advantage is that the convergence is stable and the global optimal solution can be found.
stochastic gradient descent
Stochastic gradient descent is a variant of batch gradient descent that uses only one sample in each iteration to compute gradients and update parameters. Since only one sample is used for each iteration, stochastic gradient descent is computationally fast and is especially suitable for large-scale datasets. However, due to the gradient calculation using a single sample, the parameter update of stochastic gradient descent will produce large jitter, resulting in unstable value of the objective function. To overcome this problem, a learning rate scheduling strategy can be introduced to gradually reduce the learning rate to reduce jitter and accelerate convergence.
Mini-batch gradient descent
Mini-batch gradient descent is a compromise between batch gradient descent and stochastic gradient descent. In each iteration, it uses a small set of samples (called a mini-batch) to compute gradients and update parameters. Mini-batch gradient descent combines the stability of batch gradient descent with the computational efficiency of stochastic gradient descent and is generally the most commonly used variant of gradient descent in practical applications.
Convergence
An important property of the gradient descent algorithm is convergence, that is, when the learning rate and the number of iterations are large enough, the gradient descent algorithm can converge to the minimum value of the objective function. However, in practical applications, convergence is not always guaranteed. Some objective functions may have local minima or saddle points, so that gradient descent may fall into local minima and fail to reach the global optimal solution. To deal with this situation, different optimization techniques can be used, such as random initialization parameters, adaptive learning rate and momentum, etc.
Applications of Gradient Descent in Machine Learning
The gradient descent algorithm has a wide range of applications in machine learning, especially in the process of training models. Various models such as linear regression, logistic regression, support vector machines, neural networks, etc. can use gradient descent to optimize parameters. In the field of deep learning, variants of gradient descent such as stochastic gradient descent, mini-batch gradient descent, and momentum are widely used to train deep neural networks.
Summarize
Gradient descent is an important optimization algorithm for finding the minimum or maximum of a function. It makes the value of the objective function gradually approach the optimal solution by continuously adjusting the value of the parameters. In machine learning and deep learning, gradient descent is widely used in the training process of the model to optimize the model parameters by minimizing the loss function. Different variants of gradient descent such as batch gradient descent, stochastic gradient descent, and mini-batch gradient descent have their own advantages in practical applications. In order to ensure the effect of the gradient descent algorithm, it is necessary to select an appropriate learning rate, number of iterations and optimization techniques. The application of gradient descent involves various machine learning models and deep learning models, and plays an indispensable role in practical problems.