gradient descent

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Linear Regression Blog Post

Gradient descent



Introduction

In our last article, we learned about the importance of the loss function and how lower loss values indicate better parameter values. Now, we will address the question of how to obtain these optimal parameters that minimize the loss. In this article, we will introduce the gradient descent algorithm, which will provide us with the solution.

Imagine you are standing on a mountain and your goal is to reach the lowest point in the valley. You cannot see the entire landscape, but you can feel the slope of the ground beneath your feet. Gradient descent works in a similar way.

Gradient descent allows us to find the optimal values for the model's parameters by iteratively adjusting them in the direction that decreases the loss. It relies on the concept of gradients, which represent the slopes of the loss function with respect to each parameter.

By iteratively descending along the gradients, we gradually approach the minimum of the loss function, similar to descending the mountain slope towards the valley's lowest point.
Gradient descent enables us to optimize complex models with numerous parameters efficiently. It is a fundamental technique that underlies many advanced machine learning algorithms, including deep learning. By leveraging gradient descent, we can navigate the vast parameter space and fine-tune our models, improving their ability to capture the underlying patterns and make accurate predictions.


The gradient

When we have a function that depends on multiple variables, the partial derivative measures how the function changes with respect to each individual variable, holding the other variables constant. It gives us insight into the sensitivity of the function to changes in each variable.

For a function \(f(x_1, x_2, ..., x_n)\), the partial derivative with respect to \(x_i\) is denoted as \(\frac{{\partial f}}{{\partial x_i}}\). It quantifies the rate of change of \(f\) concerning \(x_i\) while keeping the other variables fixed.

Now, let's consider a situation where we have a multivariable function \(f\) and we want to find the direction of the steepest ascent or descent. This is where the gradient comes into play.

The gradient is a vector that contains the partial derivatives of the function with respect to each variable. It provides us with the direction of the greatest rate of change of the function.

For a function \(f(x_1, x_2, ..., x_n)\), the gradient is represented as \(\nabla f = \left[\frac{{\partial f}}{{\partial x_1}}, \frac{{\partial f}}{{\partial x_2}}, ..., \frac{{\partial f}}{{\partial x_n}}\right]\).

The gradient vector points in the direction of the steepest ascent, where the function increases the most. To find the direction of the steepest descent, we take the negative of the gradient vector.

By calculating the gradient, we obtain a vector that guides us toward the most significant rate of change of the function with respect to each variable. This information is valuable for optimization algorithms like gradient descent.

Gradient descent utilizes the gradient to iteratively update the parameters of a model in the direction that minimizes the loss function. By adjusting the parameters based on the gradients, we progressively move towards the minimum of the loss function, effectively optimizing the model.

In summary, the gradient is a vector that contains the partial derivatives of a multivariable function. It provides us with the direction of the steepest ascent or descent. By leveraging the gradient in optimization algorithms like gradient descent, we can efficiently adjust the parameters to minimize the loss and improve our models.

In the next visualization try drag the red point in the parameter space and observe that the gradient do really point in the direction of steepest ascent





Gradient Descent Steps

So now we can talk about the algoritm itself.

  • Step 1: Initialization

    We start by initializing the parameters of our model with some initial values. These can be random or based on prior knowledge. For example, in linear regression, we initialize the slope and intercept parameters.

  • Step 2: Computing the Loss

    Using the current parameter values, we compute the loss function.

  • Step 3: Computing the Gradients

    Next, we compute the partial derivative of the loss function with respect to each parameter and get the gradient. As we saw, the gradients indicate the direction and magnitude of the steepest ascent of the loss function. We need to move in the opposite direction to descend towards the minimum.

  • Step 4: Updating the Parameters

    Based on the gradients, we update the parameters by taking a small step in the opposite direction of the gradients. The step size is determined by the learning rate, which controls the magnitude of parameter updates at each iteration. A smaller learning rate ensures more cautious and precise updates.

  • Step 5: Iteration

    We repeat Steps 2-4 for a certain number of iterations or until a convergence criterion is met. In each iteration, we compute the loss, gradients, and update the parameters. This process continues until we reach a point where further adjustments no longer significantly reduce the loss or until convergence is achieved.

  • Step 6: Final Parameters

    After the iterations, we obtain the final parameter values that minimize the loss function. These parameters represent the best-fit values that align the model with the data.



Final visualization

So here we came to the final visualization on linear regression. This visualization has it all: data, the best fit line, the loss function, gradients, gradient descent, and the parameters that control gradient descent.

In the upper-left window, you have sliders that control the parameters of the training: the current step and the learning rate. Feel free to experiment with different learning rates and observe their impact. Also, try different initializations by changing the location of the parameters point in the parameter space.

Additionally, try arranging the data points differently, for example, in a non-linear way, and observe how it impacts the loss function and, by extension, the training and the minimum value.



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