Gradient Descent + Matrix Determinants

Objective of the post is as follows :

1. Understand Gradient Descent
2. Awesome application of gradient descent to create matrix out of an integer such that determinant of generated matrix is same as given integer.

What is Gradient Descent ?

Gradient descent is an optimization algorithm commonly used in machine learning and mathematical optimization to minimize a function iteratively. It is an iterative method that aims to find the minimum of a function by following the steepest descent direction.

The general idea behind gradient descent is to start with an initial guess for the optimal solution and then update the guess iteratively by taking steps proportional to the negative gradient of the function at that point. The negative gradient points in the direction of the steepest descent, meaning it is the direction in which the function decreases the fastest.

Here's a simplified description of the gradient descent algorithm:

1. Choose an initial guess for the optimal solution.
2. Calculate the gradient of the function at the current guess point.
3. Update the guess by taking a step in the opposite direction of the gradient, scaled by a learning rate parameter.
4. Repeat steps 2 and 3 until a stopping criterion is met (e.g., reaching a maximum number of iterations or the change in the function value becoming sufficiently small).

The learning rate determines the step size taken at each iteration. It is a hyperparameter that needs to be carefully chosen, as a large learning rate can cause the algorithm to overshoot the minimum, while a small learning rate can lead to slow convergence.

Gradient descent is widely used in various machine learning algorithms, such as linear regression, logistic regression, and neural networks, to optimize the model parameters based on training data. In these cases, the function being minimized is typically a cost or loss function that quantifies the difference between the predicted output of the model and the true output.

There are different variants of gradient descent, such as batch gradient descent, stochastic gradient descent (SGD), and mini-batch gradient descent. Batch gradient descent calculates the gradient of the entire dataset at each iteration, which can be computationally expensive for large datasets. SGD and mini-batch gradient descent, on the other hand, use randomly selected subsets of the data (single instances or small batches) to estimate the gradient, resulting in faster iterations but potentially with more noise in the estimated gradient.

Does gradient descent always produces same results ?

No, gradient descent does not always produce the same result. The specific outcome of the gradient descent algorithm can vary based on several factors, including the initial guess, the learning rate, the stopping criterion, and the characteristics of the function being optimized.

Here are some reasons why gradient descent may not always produce the same result:

1. Initialization: The initial guess for the optimal solution can significantly affect the convergence of gradient descent. Starting from different initial points may lead to different local minima or convergence rates.
2. Learning rate: The learning rate determines the step size taken at each iteration. Choosing an inappropriate learning rate can result in overshooting or undershooting the minimum, causing the algorithm to fail to converge or converge slowly.
3. Stopping criterion: The stopping criterion determines when to terminate the iterations. Different stopping criteria, such as a maximum number of iterations or a threshold for the change in the function value, can lead to different stopping points and therefore different final results.
4. Non-convex functions: Gradient descent may encounter non-convex functions with multiple local minima. The specific local minimum reached by the algorithm can depend on the starting point and the optimization path it follows.
5. Noise and randomness: In the case of stochastic gradient descent (SGD) or mini-batch gradient descent, where only a subset of the data is used to estimate the gradient, there can be randomness in the estimated gradient due to the random selection of data points or batches. This randomness can lead to slightly different optimization paths and final results across different runs.

To mitigate some of these issues, techniques such as random initialization, learning rate schedules, adaptive learning rates, and regularization methods are often used in practice to improve the convergence and stability of gradient descent. Additionally, more advanced optimization algorithms, like Adam, RMSprop, or conjugate gradient, are available as alternatives to standard gradient descent, offering improved convergence properties in some scenarios.

Code Please !

The following code creates 3x3 matrices using gradient descent ; Copy this to any python interpreter, either offline or online one . Make a first run check the matrix produced for first time ; Hit run again , Magical , isn't it !!

Remeber this !! we'll use this to explain awesome Generative AI concepts !! Keep the excitement on for day+=1 !! Sign off :)

import numpy as np
import scipy.optimize as opt


def objective_function(matrix, desired_determinant):
? ? current_determinant = np.linalg.det(matrix)
? ? return abs(current_determinant - desired_determinant)


def construct_matrix(desired_determinant, learning_rate, num_iterations):
? ? # Initialize the matrix
? ? matrix = np.random.rand(3, 3)


? ? # Define the objective function with the desired determinant
? ? objective = lambda x: objective_function(x.reshape((3, 3)), desired_determinant)


? ? # Perform gradient descent optimization
? ? for _ in range(num_iterations):
? ? ? ? gradient = opt.approx_fprime(matrix.flatten(), objective, epsilon=1e-8)
? ? ? ? matrix -= learning_rate * gradient.reshape(matrix.shape)


? ? return matrix


# Define the desired determinant
desired_determinant = 5


# Set the learning rate and number of iterations for gradient descent
learning_rate = 0.01
num_iterations = 1000


# Construct the matrix using gradient descent
matrix = construct_matrix(desired_determinant, learning_rate, num_iterations)


# Print the matrix and its determinant
print("Matrix:")
print(matrix)
print("Determinant:", np.linalg.det(matrix))