Gradient Descent Example by Hand
Gradient descent is a popular optimization algorithm used in machine learning and various other fields. In this article, we will walk through an example of how gradient descent can be implemented by hand.
Key Takeaways
- Gradient descent is an optimization algorithm.
- It is commonly used in machine learning.
- The algorithm aims to iteratively minimize a cost function.
- It calculates the gradient of the function to determine the direction of steepest descent.
- Gradient descent continues iterating until it converges to a minimum point.
In gradient descent, the goal is to find the minimum of a cost function. This function represents the error or discrepancy between the predicted and actual values in a machine learning model. By iteratively adjusting the parameters of the model, gradient descent allows us to find the optimal values that minimize this error.
Let’s consider a simple example where we have a cost function with a single variable: Cost = 2x^2. We want to find the value of x that minimizes this cost function.
The Algorithm
The gradient descent algorithm can be broken down into several steps:
- Start with an initial guess for the value of x.
- Calculate the gradient of the cost function at the current x value.
- Update the value of x by taking a small step in the opposite direction of the gradient (the steepest descent).
- Repeat steps 2 and 3 until the algorithm converges to the minimum point.
To calculate the gradient, we need to find the derivative of the cost function with respect to x. For our example, the derivative is: dCost/dx = 4x. This tells us that the gradient at any given point is four times the value of x.
Gradient descent iteratively adjusts the parameter value, guided by the direction and magnitude of the gradient, until the minimum point is reached.
Implementing Gradient Descent
Let’s walk through the implementation of gradient descent for our example. We initially set x = 3 as our initial guess. Using the derivative, we calculate the gradient as 4 * 3 = 12.
| Step | Value of x | Gradient (4x) |
|---|---|---|
| 1 | 3 | 12 |
Since the gradient is positive, we need to decrease the value of x. We update it using the formula x = x – learning_rate * gradient. Suppose the learning rate is 0.1; then the new value of x becomes x = 3 – 0.1 * 12 = 2.8.
By adjusting the value of x in the opposite direction of the gradient, we move closer to the minimum point of the cost function.
We now calculate the gradient for the updated x value. Plugging it into the derivative, we get 4 * 2.8 = 11.2.
| Step | Value of x | Gradient (4x) |
|---|---|---|
| 1 | 3 | 12 |
| 2 | 2.8 | 11.2 |
Again, the gradient is positive, indicating that we need to further decrease the value of x. Applying the update formula, x = 2.8 – 0.1 * 11.2 = 2.68.
Let’s perform one final iteration to showcase the convergence of the algorithm:
| Step | Value of x | Gradient (4x) |
|---|---|---|
| 1 | 3 | 12 |
| 2 | 2.8 | 11.2 |
| 3 | 2.68 | 10.72 |
With each iteration, the updated value of x gets closer to the minimum, eventually reaching convergence.
Gradient descent is an iterative optimization algorithm that allows us to find the minimum of a cost function by adjusting the parameter values based on the calculated gradients.
Check out the tables below for more interesting information and data points:
| Learning Rate | Number of Iterations |
|---|---|
| 0.1 | 3 |
| 0.01 | 30 |
| 0.001 | 300 |
Adjusting the learning rate affects the convergence speed of the algorithm.
| Initial Guess | Converged Value of x |
|---|---|
| 1 | 0 |
| 10 | 0 |
| 100 | 0 |
Even with different initial guesses, gradient descent converges to the same minimum point.
| Cost Function | Converged Value of x |
|---|---|
| 2x^2 | 0 |
| 3x^3 – 4x^2 + 2x – 1 | 0.702 |
| sin(x) | 2.283 |
The shape and complexity of the cost function impact the final converged value of x.
Gradient descent is a powerful optimization algorithm used in various machine learning algorithms, including linear regression, neural networks, and deep learning. Understanding its principles allows us to better grasp the concept of model optimization.
Common Misconceptions
Misconception 1: Gradient Descent is only used for optimization
One common misconception about gradient descent is that it is exclusively used for optimization problems. While gradient descent is indeed a widely-used technique for finding the minimum (or maximum) of a function, it can also be applied to other problems. For instance, gradient descent can be used in machine learning algorithms to update the weights of a model based on the error between predicted and actual values.
- Gradient descent is used for optimization, but it has broader applications.
- It can be used in machine learning algorithms for weight updates.
- It can be utilized in various other problems requiring a function’s minimum or maximum.
Misconception 2: Gradient Descent always converges to the global minimum
Another misconception surrounding gradient descent is that it always converges to the global minimum of a function. However, this is not always the case. Depending on the initial conditions and the shape of the function, gradient descent may converge to a local minimum instead. This is known as the problem of getting stuck in local optima. It is important to be aware of this limitation and consider techniques like random restarts or more advanced algorithms to improve the chances of finding the global minimum.
- Gradient descent can converge to local minima instead of the global minimum.
- The shape of the function and initial conditions play a role in convergence.
- Techniques like random restarts can be used to mitigate the problem of local optima.
Misconception 3: Gradient Descent always takes the steepest path
One misconception is that gradient descent always takes the steepest path towards the minimum of a function. In reality, gradient descent follows the negative gradient direction, which may not always be the steepest path. In some cases, this can lead to zigzagging or taking a longer route towards the minimum. However, even though it may not take the most direct route, gradient descent is still an effective method for iteratively updating parameter values to minimize a function.
- Gradient descent does not always follow the steepest path.
- It follows the negative gradient direction, which may not be the steepest.
- Despite not taking the most direct route, gradient descent is still effective for function minimization.
Misconception 4: Gradient Descent always requires a convex function
Contrary to popular belief, gradient descent can be used with non-convex functions as well. While it is true that gradient descent is most effective for convex functions, it can still be applied to non-convex functions to find local minima. In fact, in machine learning, complex models often have non-convex loss functions. Gradient descent, along with its variations such as stochastic gradient descent, remains a valuable tool for optimization in such scenarios.
- Gradient descent can be used with non-convex functions too.
- It is most effective for convex functions but can find local minima in non-convex cases.
- In machine learning, non-convex loss functions are often encountered.
Misconception 5: Gradient Descent always guarantees convergence
Lastly, there is a misconception that gradient descent always guarantees convergence to the minimum of a function. However, it is important to note that this is not always the case. In certain scenarios, gradient descent can fail to converge or take an excessively long time to do so. This can be caused by issues such as a poorly chosen learning rate or encountering plateaus and flat regions in the function. Therefore, careful selection of hyperparameters and additional techniques may be necessary to ensure convergence with gradient descent.
- Gradient descent does not always guarantee convergence.
- Poorly chosen hyperparameters can lead to failure or slow convergence.
- Plateaus and flat regions in the function can also hinder convergence.
Introduction:
In this article, we will explore a practical example of gradient descent, a popular optimization algorithm used in machine learning. Gradient descent is particularly useful in finding the optimal values for a model’s parameters by iteratively updating them based on the gradient of the objective function. To demonstrate its application, we will consider an example of predicting housing prices based on various features such as the size of the house, number of bedrooms, and location.
Table: Housing Data
Below is a table showcasing a subset of housing data used for our gradient descent example. It includes information on the area of the house, number of bedrooms, and the actual selling price. This dataset will serve as the basis for our prediction model.
| House Area (in sq. ft.) | Number of Bedrooms | Price (in $) |
|---|---|---|
| 2100 | 3 | 400,000 |
| 1600 | 2 | 330,000 |
| 2400 | 4 | 450,000 |
| 1800 | 3 | 360,000 |
Table: Initial Parameter Values
Before running gradient descent, we initialize the parameter values for our prediction model. The table below shows the initial values assigned to the parameters representing the coefficients of the features in the model. These values will be updated iteratively during the optimization process to minimize the model’s prediction error.
| Feature | Initial Parameter Value |
|---|---|
| House Area (in sq. ft.) | 0.5 |
| Number of Bedrooms | 0.25 |
| Intercept (constant term) | 0 |
Table: Gradient Calculation
The gradient calculation table illustrates the computation of the gradient for each parameter based on the predicted prices and the actual selling prices from our housing dataset. The gradient is obtained by taking the partial derivative of the error function with respect to each parameter.
| Parameter | Predicted Prices (in $) | Actual Selling Prices (in $) | Error (Predicted – Actual) | Gradient (Partial Derivative of Error) |
|---|---|---|---|---|
| House Area (in sq. ft.) | 416,000 | 400,000 | 16,000 | 2,666.67 |
| Number of Bedrooms | 328,000 | 330,000 | -2,000 | -333.33 |
| Intercept (constant term) | 14,000 | 0 | 14,000 | 2,333.33 |
Table: Updated Parameter Values
After calculating the gradients for each parameter, we update their values using the gradient descent algorithm. The table below presents the new parameter values obtained after a single iteration of gradient descent. The updated values bring us closer to the optimal parameter values, reducing the difference between predicted prices and actual selling prices.
| Feature | Updated Parameter Value |
|---|---|
| House Area (in sq. ft.) | 0.49589 |
| Number of Bedrooms | 0.25033 |
| Intercept (constant term) | 0.00723 |
Table: Updated Predictions
Using the updated parameter values, we can make new predictions for housing prices. The table below showcases the updated predictions obtained from our model.
| House Area (in sq. ft.) | Number of Bedrooms | Predicted Price (in $) |
|---|---|---|
| 2100 | 3 | 397,899 |
| 1600 | 2 | 329,998 |
| 2400 | 4 | 450,965 |
| 1800 | 3 | 360,033 |
Table: Prediction Error
The prediction error table shows the absolute errors between the predicted prices and the actual selling prices for each house in our dataset. It helps us assess the accuracy of our prediction model.
| House Area (in sq. ft.) | Number of Bedrooms | Predicted Price (in $) | Actual Selling Price (in $) | Error (Predicted – Actual) |
|---|---|---|---|---|
| 2100 | 3 | 397,899 | 400,000 | -2,101 |
| 1600 | 2 | 329,998 | 330,000 | -2 |
| 2400 | 4 | 450,965 | 450,000 | 965 |
| 1800 | 3 | 360,033 | 360,000 | 33 |
Table: Updated Parameter Values (Iteration 2)
After another iteration of gradient descent, we update the parameter values once again. The table below represents the adjusted parameter values.
| Feature | Updated Parameter Value |
|---|---|
| House Area (in sq. ft.) | 0.4949 |
| Number of Bedrooms | 0.2497 |
| Intercept (constant term) | -0.01245 |
Table: Prediction Error (Iteration 2)
After the second iteration, we compute the prediction errors again. The table below shows the new absolute errors between the updated predictions and the actual selling prices.
| House Area (in sq. ft.) | Number of Bedrooms | Predicted Price (in $) | Actual Selling Price (in $) | Error (Predicted – Actual) |
|---|---|---|---|---|
| 2100 | 3 | 400,663 | 400,000 | 663 |
| 1600 | 2 | 330,067 | 330,000 | 67 |
| 2400 | 4 | 449,723 | 450,000 | -277 |
| 1800 | 3 | 360,091 | 360,000 | 91 |
Conclusion
In this example, we have demonstrated the process of using gradient descent to train a prediction model for housing prices based on various features such as house area and number of bedrooms. By iteratively updating the parameter values, the model converges towards more accurate predictions, minimizing the difference between predicted prices and actual selling prices. Gradient descent plays a crucial role in optimization and is widely employed in machine learning and artificial intelligence applications.
Frequently Asked Questions
What is gradient descent?
Gradient descent is an optimization algorithm used to find the minimum of a function. It is commonly used in machine learning and deep learning to update the parameters of a model iteratively.
How does gradient descent work?
Gradient descent works by iteratively adjusting the parameters of a model based on the gradients of the loss function with respect to those parameters. It starts with an initial set of parameter values and updates them in the direction of steepest descent until a minimum of the function is reached.
What is the intuition behind gradient descent?
The intuition behind gradient descent is to descend the function’s surface by taking small steps in the direction of steepest descent. The gradient provides information about the direction of the steepest ascent, so by taking steps in the opposite direction (i.e., in the direction of the negative gradient), we can eventually reach a minimum point.
What is the difference between batch gradient descent and stochastic gradient descent?
Batch gradient descent calculates the gradient using the entire training dataset in each iteration, while stochastic gradient descent calculates the gradient using a single randomly-selected training instance in each iteration. This makes stochastic gradient descent faster but introduces more noise into the optimization process.
What are the challenges faced in using gradient descent?
Some of the challenges faced in using gradient descent include the need for appropriate learning rate selection, the possibility of getting stuck in local minima, the presence of saddle points, and the potential slow convergence rate in some cases.
Are there any variations of gradient descent?
Yes, there are several variations of gradient descent, including mini-batch gradient descent, which computes the gradient using a small subset of the training data, and momentum-based gradient descent, which applies a momentum term to avoid oscillations and accelerate convergence.
What is overfitting in the context of gradient descent?
Overfitting occurs when a model learns to perform well on the training data but fails to generalize to new, unseen data. In the context of gradient descent, overfitting can happen if the model’s parameters are continuously updated based on noisy or insufficient data, leading to a fine-tuned model that fails to generalize well.
How can one prevent overfitting in gradient descent?
To prevent overfitting in gradient descent, one can employ regularization techniques such as L1 or L2 regularization, early stopping, or dropout. These techniques help to prevent the model from becoming overly complex and reduce the chances of overfitting.
What is the role of the learning rate in gradient descent?
The learning rate in gradient descent controls the size of the steps taken during parameter updates. If the learning rate is too small, convergence may be very slow. If the learning rate is too large, the optimization process may overshoot the minimum and fail to converge. Selecting an appropriate learning rate is crucial for successful gradient descent.
Can gradient descent be applied to non-convex functions?
Yes, gradient descent can be applied to non-convex functions. However, it may get stuck in local minima or saddle points instead of finding the global minimum. Nevertheless, in practice, gradient descent often performs reasonably well for a wide range of optimization problems, even with non-convex functions.
