Detecting Global Optimum Convergence

Can we apply elementary row operations on weights matrix to check if the we have reached global optimum solution ?

We know that by applying elementary row operations on a system of linear equations doesn't change the solution of the system. In machine learning pursuit if indeed we have reached a global optimum solution then can we apply a random elementary row operation on weights and check if the solution doesn't change to detect if the we have reached global optimum solution ?

Let's try this with an experiment :

Experiment Setup:

1. Generate synthetic data for linear regression.
2. Train a linear regression model using gradient descent.
3. Apply elementary row operations on the weight matrix.
4. Evaluate the loss before and after applying the row operations.

import numpy as np
import matplotlib.pyplot as plt


# Generate synthetic data
np.random.seed(0)
X = 2 * np.random.rand(100, 1)
y = 4 + 3 * X + np.random.randn(100, 1)


# Add bias term to X
X_bias = np.c_[np.ones((100, 1)), X]


# Initialize weights
weights = np.random.randn(2, 1)


# Hyperparameters
learning_rate = 0.01
n_iterations = 1000


# Train linear regression model using gradient descent
for iteration in range(n_iterations):
? ? gradients = 2 / 100 * X_bias.T.dot(X_bias.dot(weights) - y)
? ? weights -= learning_rate * gradients


# Calculate loss before row operations
loss_before = np.mean((X_bias.dot(weights) - y)**2)


# Apply elementary row operations (swap rows)
weights_row_op = np.array([weights[1], weights[0]])


# Calculate loss after row operations
loss_after = np.mean((X_bias.dot(weights_row_op) - y)**2)


print(f"Loss before row operations: {loss_before}")
print(f"Loss after row operations: {loss_after}")


# Plot
plt.scatter(X, y)
plt.plot(X, X_bias.dot(weights), label='Before Row Operations')
plt.plot(X, X_bias.dot(weights_row_op), label='After Row Operations')
plt.legend()
plt.show()        

Expected Output:

You'll notice that the loss before and after applying the row operations will be different, and the model fit will also be different. This demonstrates that elementary row operations are not suitable for checking if a global optimum has been reached.

Run this code to see how the loss changes and how the fit to the data changes after applying elementary row operations. This should empirically demonstrate that such operations are not useful for optimization in machine learning.

Elementary row operations are a set of operations used primarily in linear algebra to manipulate matrices, particularly to solve systems of linear equations. These operations include:

1. Swapping two rows
2. Multiplying a row by a non-zero scalar
3. Adding or subtracting the multiple of one row to another row

In the context of machine learning and optimization, the weight matrix is typically optimized using gradient-based methods like stochastic gradient descent (SGD) or its variants (e.g., Adam, RMSprop). The goal is to minimize a loss function, which measures the difference between the predicted and actual outputs.

Why Elementary Row Operations are Not Suitable for Checking Global Optimum:

1. Non-Linearity: Many machine learning models, especially neural networks, are non-linear. Elementary row operations are linear transformations, so they won't capture the complexity of the optimization landscape.
2. Loss Function: The optimization process aims to minimize a loss function, which is a function of the weights. Elementary row operations on the weight matrix do not guarantee that the loss function will reach its minimum.
3. High-Dimensional Space: The weight matrix often exists in a high-dimensional space, and the optimization landscape can have multiple local minima and maxima. Elementary row operations are not designed to navigate such complex spaces.
4. Gradient Information: Methods like SGD use gradient information to update the weights. Elementary row operations do not use this information, making them less effective for optimization.
5. Computational Efficiency: Gradient-based methods are computationally efficient and can be parallelized easily, whereas applying elementary row operations iteratively would be computationally expensive and less efficient.
6. Global vs Local Optimum: Even if you could manipulate the weight matrix to reach a local minimum, there's no guarantee that this would be a global minimum, especially for non-convex loss functions.

In summary, elementary row operations are not suitable for checking if you have reached the global optimum of a machine learning model. Instead, techniques like monitoring the loss function, early stopping, or using specialized optimization algorithms are more appropriate for this purpose.