Gradient Descent
The gradient method is one of the widely used methods in machine learning and optimization algorithms, the main objective of which is to iteratively update parameters in order to find the minimum (or maximum) value of a function.
In machine learning, the goal is usually to minimize the cost function (also called loss function). For example, in regression and classification problems, a cost function is defined to represent the error between predicted and actual values, and it helps to find the parameter values that minimize this cost function.
The basic idea of the gradient method is to calculate the gradient (derivative) of the cost function from the current parameter values, and then update the parameters by a certain step size (called the learning rate) in the direction of that gradient. By repeating this update, it is expected that the parameters will gradually converge to the optimal ones.
There are several variations of the gradient method, the main ones being
- Batch Gradient Descent: The parameters are updated by computing the gradient using all the training data at each update step. Computational cost can be high due to the large amount of data handled at one time.
- Stochastic Gradient Descent (SGD): Calculates the gradient and updates the parameters using a single randomly selected data point at each update step. Due to the random nature of the data, the computational cost is lower, but convergence may be unstable.
- Mini-batch Gradient Descent: An intermediate method between batch gradient descent and stochastic gradient descent, where some data points (mini-batches) are randomly selected to compute the gradient and update the parameters. This method strikes a balance between computational efficiency and stability.
The gradient method is a fundamental component of machine learning algorithms and is widely used for training neural networks. However, there are some caveats, such as the selection of an appropriate learning rate and the possibility of convergence to a locally optimal solution. To address those issues, a number of derived algorithms and optimization methods have been proposed.
Mathematical model of the gradient method
The gradient method is a method to search for the minimum (or maximum) value of a function using a mathematical model. Here, we describe the Batch Gradient Descent method, which is commonly used as a mathematical model.
First, the problem to be considered is to define an objective function (cost function or loss function) that represents the function to be optimized. Let J(θ) denote the objective function (where J is the objective function and θ is a vector of parameters (weights, biases, etc.) to be optimized), the batch gradient descent method updates the parameters in each step as follows.
\[\theta:=\theta-\alpha\cdot\nabla J(\theta)\]
where α is a constant called the learning rate, which controls the step size of the update. ∇J(θ) is the gradient vector for the objective function J(θ), and is a vector whose elements are partial derivatives for each parameter. Specifically, ∇J(θ) is calculated as follows.
\[\nabla J(\theta)=\left[\frac{\partial J(\theta)}{\partial\theta_1},\frac{\partial J(\theta)}{\partial\theta_2},\dots,\frac{\partial J(\theta)}{\partial\theta_n}\right]\]
where n represents the number of parameters, and each partial derivative is obtained by computing the gradient of the objective function for that parameter.
By repeating the above update equation, it is expected that the parameter θ will converge toward the minimum value of the objective function J(θ), but it was necessary to select an appropriate learning rate and to prevent convergence to a locally optimal solution. In contrast, the algorithms introduced earlier, such as Momentum, AdaGrad, RMSprop, and Adam, were developed to improve the convergence rate of the gradient method and to improve convergence.
Algorithms used in the gradient method
Various derivative algorithms exist for gradient methods. These algorithms are usually aimed at improving the convergence speed of the gradient method or making it less prone to local optimum solutions. The following describes some commonly used gradient derivation algorithms.
- Momentum: Momentum is a method that smoothes the update direction by taking into account information from previous gradients. This allows for faster updates while reducing local oscillations. Momentum is so named because it resembles the concept of momentum in physics.
- AdaGrad (Adaptive Gradient Algorithm): AdaGrad is a method that automatically adjusts the learning rate for each parameter update using past gradient information. It improves learning convergence by decreasing the learning rate for frequently used parameters and increasing the learning rate for parameters that are updated infrequently.
- RMSprop (Root Mean Square Propagation): RMSprop is an improved version of AdaGrad that improves convergence by effectively retaining past gradient information using exponential moving average.
- Adam (Adaptive Moment Estimation): Adam is a combination of Momentum and RMSprop that simultaneously adjusts the learning rate and estimates moments of past gradients. This improves the adaptability and convergence of the learning rate.
- Adadelta (Adaptive Delta): Adadelta is an extension of RMSprop that performs a moving average of past gradients and automatic adjustment of the learning rate; Adadelta does not need to keep learning rate values, thus reducing memory usage.
These algorithms are widely used in neural network training, for example, and can achieve more effective convergence for a variety of problems. However, since the optimal algorithm depends on the problem, trial and error is required in actual use.
The details of each algorithm are described below.
Momentum Algorithm
The momentum algorithm is a type of gradient method that accelerates convergence by adding inertia in the direction of the gradient during training. This method can result in faster convergence compared to the usual batch gradient descent method.
In the momentum algorithm, the parameter update formula is as follows.
\begin{eqnarray}& &v_t=\beta\cdot v_{t-1}+(1-\beta)\cdot\nabla J(\theta_t)\\& &\theta_{t+1}=\theta_t-\alpha\cdot v_t\end{eqnarray}
Here is the following.
- vt: Momentum (velocity) vector at time step t
- β: coefficient of momentum (usually 0.9 is used)
- ∇J(θt): gradient vector for the objective function J(θ) at time step t
- θt: value of the parameter at time step t
- α: learning rate
In the momentum algorithm, each step combines the gradient information with the momentum of the previous step to give speed in the gradient direction. This smoothes the direction of the update and makes it easier to get out of the local optimum solution.
For example, when entering a valley, even if the gradient is in the forward direction, the velocity vector from the previous step works in the opposite direction, allowing the user to overcome the valley, and after exiting the valley, the velocity vector is accumulated, allowing the user to take a larger step.
The momentum algorithm is particularly effective in avoiding locally optimal solutions and is widely used in training neural networks. However, the selection of the appropriate momentum coefficient and learning rate is important, and these values need to be adjusted depending on the problem.
AdaGrad(Adaptive Gradient Algorithm)
AdaGrad is a type of machine learning optimization algorithm that automatically adjusts the learning rate, aiming to improve convergence by using a large learning rate in the early stages of learning and reducing the learning rate as learning progresses.
AdaGrad accumulates past gradient information for each parameter and adjusts the learning rate based on that gradient information. Specifically, for each parameter θi, it keeps the accumulated squared gradient information up to the previous time t. This is denoted as Gt,i.
The update formula is as follows.
\begin{eqnarray}G_{t,i}&=&G_{t-1,i}+(\nabla J(\theta_{t,i}))^2\\\theta_{t+1,i}&=&\theta_{t,i}-\frac{\alpha}{\sqrt{G_{t,i}+\epsilon}}\cdot\nabla J(\theta_{t,i})\end{eqnarray}
where the following is obtained.
- Gt,i : accumulation of squared gradient information up to time t
- ∇J(θt,i) : gradient for θi with respect to the objective function J(θ) at time t
- α : learning rate
- ε : small value to avoid dividing by zero (usually 10-8, etc.)
AdaGrad is characterized by the fact that it accumulates the gradient information for each parameter as a sum of squares, making the learning rate smaller in the direction where the past gradient is large and larger in the direction where the past gradient is small. This suppresses learning in steep directions and promotes learning in flat directions, thus increasing the likelihood of stable convergence.
However, AdaGrad also has some problems. As learning progresses, the sum of squares of the gradients increases, and the learning rate rapidly decreases, making it unsuitable for long-term learning. Therefore, later proposed methods, such as RMSprop and Adam, have improved AdaGrad to make it more convergent and efficient.
RMSprop(Root Mean Square Propagation)
RMSprop is an optimization algorithm for machine learning and an improved version of AdaGrad. rmsprop automatically adjusts the learning rate to keep it appropriate and effective.
RMSprop maintains a moving average of the squares of past gradient information for each parameter. This alleviates the problem with AdaGrad of rapidly decreasing learning rates and allows for more stable learning.
The update formula is as follows.
\begin{eqnarray}G_{t,i}&=&\beta\cdot G_{t-1,i}+(1-\beta)\cdot (\nabla J(\theta_{t,i}))^2\\\theta_{t+1,i}&=&\theta_{t,i}-\frac{\alpha}{\sqrt{G_{t,i}+\epsilon}}\cdot\nabla J(\theta_{t,i})\end{eqnarray}
where the following is obtained.
- Gt,i : Cumulative squared gradient information up to time t
- β : coefficient of moving average (usually 0.9 is used)
- ∇J(θt,i) : gradient for θi with respect to the objective function J(θ) at time t
- α : learning rate
- ε : small value to avoid dividing by zero (usually 10-8, etc.)
In RMSprop, the learning rate is adjusted using a moving average of the squares of the past gradient information. The learning rate is reduced in the direction of larger gradients and increased in the direction of smaller gradients. This improves convergence while applying the appropriate learning rate to each parameter.
RMSprop is widely used as a method that solves the AdaGrad problem and has shown good performance in training neural networks. In addition, the later proposed Adam algorithm is based on RMSprop with further improvements.
Adam(Adaptive Moment Estimation)
Adam is a machine learning optimization algorithm that combines RMSprop and the momentum algorithm. It is very effective and widely used in neural network training.
Adam’s update formula is expressed as follows.
\begin{eqnarray}m_t&=&\beta_1\cdot m_{t-1}+(1-\beta_1)\cdot\nabla J(\theta_t)\\v_t&=&\beta_2\cdot v_{t-1}+(1-\beta_2)\cdot (\nabla J(\theta_t))^2\end{eqnarray}
where the following is obtained.
- mt : average of the gradient up to time t (momentum)
- vt : moving average of the square of the gradient up to time t
- β1 : Coefficient of momentum (usually 0.9 is used)
- β2 : coefficient of moving average of the square (usually a value such as 0.999 is used)
- θt : value of the parameter at time t
The update formula is then as follows
Here is the following
- \(\hat{m}_t\): bias-corrected momentum
- \(\hat{v}_t\): bias-corrected moving average of squares
- α : Learning rate
- ε : Small value to avoid dividing by 0 (usually 10-8, etc.)
In Adam, mt and vt retain past gradient information, and bias correction is applied to this information to suppress the effect of early learning and adaptively adjust the learning rate as learning progresses. This characteristic is expected to result in effective convergence in learning steep valleys and flat regions.
Adam generally performs very well in optimizing deep learning models and, as a result, is widely used in many machine learning tasks. However, the choice of hyperparameters β1, β2, α, and ϵ must be adjusted depending on the problem
Adadelta(Adaptive Delta)
Adadelta will be one of the optimization algorithms for machine learning proposed to solve the problem of RMSprop. adadelta aims to improve the efficiency and convergence of learning by increasing the adaptivity of the learning rate.
Instead of keeping a moving average of the square of the gradient as in RMSprop, Adadelta keeps a moving average of the square of the past gradient and a moving average of the square of the past updates. This alleviates the problem of rapidly decreasing learning rates and ensures that convergence is stable.
The update formula is expressed as follows.
where we have the following.
- E[g2]t: Moving average of the square of the gradient up to time t
- ρ: Coefficient of moving average of the square of the gradient (usually a value such as 0.95 is used)
- ∇J(θt): gradient of the objective function J(θ) at time t
- Δθt : amount of parameter update at time t
- E[Δθ2]t-1 : moving average of the square of the updated parameter up to time t-1
- ε : Small value to avoid dividing by 0 (usually 10-8, etc.)
In Adadelta, E[g2]t and E[Δθ2]t-1 are kept and the learning rate is adjusted using the moving average of the square of these values. The learning rate is reduced in the direction of larger gradient information and increased in the direction of smaller gradient information. By using the amount of past updates, the learning rate is automatically adjusted and convergence is expected to be stable.
Adadelta, along with RMSprop and Adam, is one of the most effective optimization algorithms and has been successful in learning deep learning models. It should be noted that appropriate hyperparameters need to be adjusted and its applicability to specific problems needs to be evaluated.
Libraries and platforms where the gradient method is available
The gradient method is a very common machine learning optimization technique and is available in many machine learning libraries and frameworks. The following are some of the main machine learning libraries and platforms where gradient methods can be used.
- TensorFlow: An open source deep learning framework developed by Google that supports a variety of optimization methods, including gradient methods.
- PyTorch: An open source deep learning framework developed by Facebook that supports gradient methods and, like TensorFlow, makes many optimization algorithms available.
- scikit-learn: a machine learning library widely used in Python that provides a variety of optimization algorithms, including gradient methods.
- Keras: A high-level neural network library that uses backends such as TensorFlow, Theano, and Microsoft Cognitive Toolkit (CNTK) and also supports gradient methods.
- MXNet: Apache MXNet is an open source deep learning framework that supports gradient methods.
- Caffe: A deep learning framework developed by Berkeley Vision and Learning Center (BVLC) that supports a variety of optimization algorithms, including gradient methods.
- Microsoft Cognitive Toolkit (CNTK): A deep learning framework developed by Microsoft that provides many optimization algorithms including gradient methods.
Application Examples of the Gradient Method
The gradient method is widely applied to machine learning and optimization problems. Specific applications are described below.
- Deep Learning (Neural Networks): In neural network training, the gradient method is used to learn optimal weights and biases. Using gradient descent and its derivative algorithms (Adam, RMSprop, etc.), the parameters of the network are adjusted and learned to minimize the objective function.
- Linear Regression: In linear regression, a linear model is learned to represent the relationship between inputs and outputs. For linear regression using the least squares method, it is common to use the gradient descent method to obtain the regression coefficients.
- Logistic Regression: In classification problems, logistic regression is used to classify data into two classes. Logistic regression uses the gradient method to learn weights and make probabilistic predictions.
- Support Vector Machines (SVM): SVMs can be powerful algorithms applied to classification and regression problems.
- Clustering: Clustering algorithms also use gradient methods to optimize cluster centers and clustering indices.
- Feature Selection: Feature selection is a method for selecting features to be used in training machine learning models, using the gradient method to evaluate the importance of features and reduce unnecessary features.
- Natural Language Processing (NLP): In NLP tasks, gradient methods are also commonly used to train neural network models and language models.
Finally, examples of gradient method implementations in various languages will be discussed.
Example implementation of the gradient method in python
As an example of the gradient method, a Python implementation of the gradient descent method (Gradient Descent) for finding the minimum of a simple function is shown. In the following example, the goal is to minimize a quadratic function.
import numpy as np
def cost_function(x):
# Quadratic function example: f(x) = x^2 + 5x + 6
return x**2 + 5*x + 6
def gradient(x):
# The gradient of a quadratic function: f'(x) = 2x + 5
return 2*x + 5
def gradient_descent(learning_rate, iterations, initial_x):
x = initial_x
for _ in range(iterations):
grad = gradient(x)
x = x - learning_rate * grad
return x
if __name__ == "__main__":
learning_rate = 0.1
iterations = 100
initial_x = 0.0
min_x = gradient_descent(learning_rate, iterations, initial_x)
min_value = cost_function(min_x)
print("Minimum x: {:.2f}".format(min_x))
print("Minimum y: {:.2f}".format(min_value))
In this example, the gradient descent method is used to find the minimum value of the quadratic function f(x)=x2+5x+6. The cost_function function defines the objective function, the gradient function calculates its gradient, and the gradient_descent function uses a specified learning rate and number of iterations to search for the minimum value from the initial value initial_x. The gradient_descent function searches for the minimum value from the initial value initial_x.
The gradient method is generally a simple algorithm that includes a part to calculate the gradient of the objective function and a part to update the parameters. In actual applications, it is often applied to more complex functions and higher dimensional data, but the basic idea is similar to the above example.
Example implementation of the gradient method with clojure
To implement the gradient method in Clojure, the following example is given. Here we show an implementation of the gradient descent method that minimizes a quadratic function.
(defn cost-function [x]
(+ (* x x) (* 5 x) 6))
(defn gradient [x]
(+ (* 2 x) 5))
(defn gradient-descent [learning-rate iterations initial-x]
(loop [x initial-x]
(if (zero? iterations)
x
(let [grad (gradient x)]
(recur (- x (* learning-rate grad)))))))
(defn -main []
(let [learning-rate 0.1
iterations 100
initial-x 0.0
min-x (gradient-descent learning-rate iterations initial-x)
min-value (cost-function min-x)]
(println (str "Minimum x: " (format "%.2f" min-x)))
(println (str "Minimum y: " (format "%.2f" min-value)))))
(-main)
This Clojure code is very similar to the Python implementation example: the cost-function function computes the objective function, the gradient function computes its gradient, and the gradient-descent function uses a specified learning rate and number of iterations to search for the minimum value from the initial value initial-x The minimum value is searched for.
Because Clojure is a type of Lisp and has features of functional programming, many implementations use recursion. In this example, loop and recur are used to iterate.
Example implementation of the gradient method in Rust
fn cost_function(x: f64) -> f64 {
x * x + 5.0 * x + 6.0
}
fn gradient(x: f64) -> f64 {
2.0 * x + 5.0
}
fn gradient_descent(learning_rate: f64, iterations: usize, initial_x: f64) -> f64 {
let mut x = initial_x;
for _ in 0..iterations {
let grad = gradient(x);
x -= learning_rate * grad;
}
x
}
fn main() {
let learning_rate = 0.1;
let iterations = 100;
let initial_x = 0.0;
let min_x = gradient_descent(learning_rate, iterations, initial_x);
let min_value = cost_function(min_x);
println!("Minimum x: {:.2}", min_x);
println!("Minimum y: {:.2}", min_value);
}
This Rust code is very similar to the Python and Clojure implementation examples: the cost_function function computes the objective function, the gradient function computes its gradient, and the gradient_descent function uses a specified learning rate and number of iterations to search for the minimum value from the initial value The minimum value is searched from initial_x.
Rust is a systems programming language, which emphasizes safety and performance. In this example, the mut keyword is used for mutable changes of variables, and the for _ in 0..iterations is used for loops.
Because Rust is a statically typed language, it can detect type and memory errors at compile time. This characteristic allows it to be used with high reliability in numerical computations such as machine learning.
Reference Information and Reference Books
For a mathematical approach to machine learning, see “Mathematics in Machine Learning” for more details.
Refernce book is “
“
“
コメント