Gradient descent is arguably the most fundamental algorithm in deep learning. Without it, deep learning as we know it would simply not exist. Despite its critical importance, the gradient descent algorithm is remarkably straightforward, relying primarily on calculus concepts like derivatives.
At its core, gradient descent is all about finding the minimum of a function that represents the errors or mistakes made by a deep learning model. Here is how it works:
Start with a random initial guess of the solution (model parameters/weights).
Compute the derivative of the error function at that guess point.
Update the guess by moving in the direction opposite to the derivative, scaled by a small learning rate.
Repeat steps 2 and 3 iteratively, getting closer to the minimum of the error function
The key idea is to use the derivative (or gradient in higher dimensions) to determine which direction updates to the parameters will minimize the errors made by the model. It is analogous to descending a hill by taking steps toward the steepest descent.
While easy to understand conceptually, there are some nuances to gradient descent in practice:
Choosing an appropriate learning rate is crucial - too large and you may overshoot; too small and convergence will be slow.
The algorithm can get stuck in local minima instead of the global minimum.
Issues like vanishing/exploding gradients can destabilize training.
It provides an approximate, not necessarily perfect, solution.
Now, let's understand the above theory points using one function's minimization ( in 1 Dimension) process using gradient descent algorithm steps in Python.
The function we have is,
y = 3x*x - 3x + 6
Our goal is to find the value of x that minimizes this function, i.e. the x-value at the lowest point of the parabola.
Now let's implement the gradient descent algorithm: (Using the steps indicated above) :
Start with a random initial guess, e.g. x = -2
Compute the derivative dy/dx = 6x - 3
At x = -2, dy/dx = 6*(-2) - 3 = -15
Update x in the opposite direction of the derivative: x_new = x - learning_rate dy/dx Let's use a learning_rate of 0.01 x_new = -2 - 0.01(-15) = -1.85
Repeat steps 2-4 with x = -1.85 until convergence.
Theoretically, to find the exact minimum value of a function, we can set its derivative equal to zero and solve for the value of the independent variable (x) at which the derivative becomes zero.
In the case of the function y = 3x^2 - 3x + 6, we can follow these steps:
Take the derivative of the function: dy/dx = 6x - 3
Set the derivative equal to zero: 6x - 3 = 0
Solve for x: 6x = 3 x = 3/6 x = 0.5
So, by setting the derivative dy/dx = 6x - 3 equal to zero and solving for x, we find that the function y = 3x^2 - 3x + 6 attains its minimum value when x = 0.5.
This analytical approach gives us the exact minimum value of the function, whereas the gradient descent algorithm provides an approximate solution by iteratively moving towards the minimum but not necessarily reaching it exactly.
In practice, the gradient descent algorithm is widely used for minimizing complex, high-dimensional functions where finding an analytical solution is intractable or impossible. However, for simple functions like the one in this example, we can find the exact minimum by setting the derivative equal to zero and solving for the independent variable.
Now, let's implement the above algorithm using Python code, visualize it with a function and its derivative graph, and get the minimum value.
Gradient Descent algorithm for 1-d Function
You will get the below output:
Converged at x=0.5000 after 268 iterations
Minimum found at x=0.5000**Note: The threshold value in above the gradient_descent function is a hyperparameter that determines how close the derivative needs to be to zero for the algorithm to consider convergence and stop iterating. The choice of the threshold value is a trade-off between accuracy and computational efficiency. Here A common default value for the threshold is taken 1e-6. (Try taking values like 0.01,0.05 etc.)
Below is the plot of the function and its derivative and the value of x at which the function
y = 3x^2 - 3x + 6 achieves its minimum value.
Output Plot
To conclude, In the world of deep learning and optimization, the gradient descent algorithm stands out as a true workhorse. Despite its conceptual simplicity, it provides a powerful way to train complex models and find optimal solutions to challenging problems.
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