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“梯度下降法”的原理

发布时间 2023-08-22 13:17:00作者: 立体风

梯度下降法是一个用于优化多变量函数的迭代方法。在深度学习和机器学习中，通常用它来优化损失函数，从而找到一个模型的最优参数。

以下是梯度下降法的原理详解：

  1. 目标：我们的目标是找到函数(f(\theta))的最小值，其中(\theta)是一个参数向量。在机器学习中，这个函数通常是损失函数或代价函数。

  2. 梯度的定义：函数在某一点的梯度表示该点的最陡上升方向。具体来说，对于函数(f(\theta))，其梯度是一个向量，其每个元素是对(\theta)的每个元素的偏导数。用数学符号表示，梯度为：

    [
    \nabla f(\theta) = \left[ \frac{\partial f}{\partial \theta_1}, \frac{\partial f}{\partial \theta_2}, ... , \frac{\partial f}{\partial \theta_n} \right]
    ]

  3. 更新规则：在每一次迭代中，我们都会按照当前点的梯度的反方向更新(\theta)，这样可以朝向函数值减小的方向移动。更新规则如下：

    [
    \theta_{\text{new}} = \theta_{\text{old}} - \alpha \times \nabla f(\theta_{\text{old}})
    ]

    其中，(\alpha) 是学习率，是一个正数，决定了每一步移动的大小。较大的学习率意味着更大的步长，但也可能导致算法不收敛；较小的学习率则可能使收敛速度变慢。

  4. 收敛：理想情况下，梯度下降会在函数的最小点处收敛，这时的梯度接近于零。但在实际应用中，我们通常设置迭代次数上限或当梯度的模小于某个阈值时停止迭代。

  5. 种类：

    • 批量梯度下降（BGD）：在每一步都使用整个数据集计算梯度。
    • 随机梯度下降（SGD）：在每一步只随机选择一个数据点来计算梯度。
    • 小批量梯度下降（MBGD）：在每一步选择一个小批量的数据来计算梯度。

  6. 局部最小与全局最小：有时函数可能有多个局部最小值。梯度下降可能会收敛到局部最小值而不是全局最小值，这取决于初始化的参数和学习率。对于某些函数，如凸函数，只有一个全局最小值，梯度下降可以保证找到这个最小值。

  7. 速度和优化：许多方法都被提出来加速梯度下降或帮助其找到更好的最小值，如动量、Adam、RMSProp等。

总的来说，梯度下降法是通过迭代地更新参数，朝着使得函数值下降的方向移动，从而找到函数的最小值。

英文版：
Gradient Descent is one of the core algorithms in machine learning and deep learning for optimizing functions, and here's an introduction:

Gradient Descent Method

Objective: To minimize a given function. In machine learning, this is typically a cost or loss function that measures how well our model is doing.

  1. Gradient: The gradient of a function at a particular point is a vector that points in the direction of the steepest increase of the function. Mathematically, for a function (f(\theta)), where (\theta) is a vector of parameters, the gradient is a vector of the first order partial derivatives with respect to each parameter.

  2. Basic Idea: To minimize a function using gradient descent, you start with an initial guess for the function's parameters and iteratively adjust the parameters to move in the direction opposite to the gradient. This "downhill" movement is expected to lead towards the function's minimum.

  3. Update Rule: At each step, the parameters are adjusted based on the formula:
    [
    \theta_{\text{new}} = \theta_{\text{old}} - \alpha \times \nabla f(\theta_{\text{old}})
    ]
    Where:

    • ( \nabla f(\theta_{\text{old}}) ) is the gradient of the function at the current parameter values.
    • ( \alpha ) is the learning rate, a positive scalar determining the step size.

  4. Learning Rate (( \alpha )): The learning rate is crucial. If it's too large, the method might overshoot the minimum and fail to converge. If it's too small, the method can be very slow. Adaptive methods can also be used to adjust the learning rate as the optimization proceeds.

  5. Variants:

    • Batch Gradient Descent: Uses the entire dataset to compute the gradient at every step.
    • Stochastic Gradient Descent (SGD): Uses a single random data point to compute the gradient at every step.
    • Mini-batch Gradient Descent: Uses a small random subset (mini-batch) of the data to compute the gradient.

  6. Challenges:

    • Local Minima: The algorithm could converge to a local minimum rather than the global minimum.
    • Saddle Points: Places where the gradient is zero but are neither a minimum nor a maximum.
    • Speed: Depending on the size of the dataset and the complexity of the function, gradient descent can be slow.

  7. Advanced Optimizers: Over the years, several optimizations over the basic gradient descent have been proposed to make it faster and more reliable. Examples include Momentum, Adagrad, RMSProp, and Adam.

In summary, Gradient Descent is an iterative optimization algorithm used to find the minimum value of a function. By continuously moving in the direction opposite to the gradient, the algorithm seeks to find the point where the function has its lowest value.