gradient descent Algorithm

Gradient descent is an iterative optimization algorithm widely used in machine learning and deep learning to minimize a given objective function. The objective function, often referred to as the loss or cost function, is a measure of how well the model is performing on a given task. The goal is to find the model parameters that minimize the value of this function. Gradient descent works by iteratively adjusting the model parameters in the direction of the steepest decrease in the objective function until convergence is reached or a stopping criterion is met. The algorithm starts with an initial set of parameters and computes the gradient of the objective function with respect to these parameters. The gradient is a vector that points in the direction of the steepest increase in the function and its magnitude indicates the rate of change. The model parameters are then updated by taking a step proportional to the negative of the gradient and the learning rate, which is a hyperparameter that determines the size of the step. This process is repeated until the gradient becomes close to zero or some maximum number of iterations have been reached. Variants of gradient descent, such as stochastic gradient descent (SGD) and mini-batch gradient descent, introduce randomness or work with a subset of the data to speed up convergence and improve generalization.
"""
Implementation of gradient descent algorithm for minimizing cost of a linear hypothesis function.
"""
import numpy

# List of input, output pairs
train_data = (
    ((5, 2, 3), 15),
    ((6, 5, 9), 25),
    ((11, 12, 13), 41),
    ((1, 1, 1), 8),
    ((11, 12, 13), 41),
)
test_data = (((515, 22, 13), 555), ((61, 35, 49), 150))
parameter_vector = [2, 4, 1, 5]
m = len(train_data)
LEARNING_RATE = 0.009


def _error(example_no, data_set="train"):
    """
    :param data_set: train data or test data
    :param example_no: example number whose error has to be checked
    :return: error in example pointed by example number.
    """
    return calculate_hypothesis_value(example_no, data_set) - output(
        example_no, data_set
    )


def _hypothesis_value(data_input_tuple):
    """
    Calculates hypothesis function value for a given input
    :param data_input_tuple: Input tuple of a particular example
    :return: Value of hypothesis function at that point.
    Note that there is an 'biased input' whose value is fixed as 1.
    It is not explicitly mentioned in input data.. But, ML hypothesis functions use it.
    So, we have to take care of it separately. Line 36 takes care of it.
    """
    hyp_val = 0
    for i in range(len(parameter_vector) - 1):
        hyp_val += data_input_tuple[i] * parameter_vector[i + 1]
    hyp_val += parameter_vector[0]
    return hyp_val


def output(example_no, data_set):
    """
    :param data_set: test data or train data
    :param example_no: example whose output is to be fetched
    :return: output for that example
    """
    if data_set == "train":
        return train_data[example_no][1]
    elif data_set == "test":
        return test_data[example_no][1]


def calculate_hypothesis_value(example_no, data_set):
    """
    Calculates hypothesis value for a given example
    :param data_set: test data or train_data
    :param example_no: example whose hypothesis value is to be calculated
    :return: hypothesis value for that example
    """
    if data_set == "train":
        return _hypothesis_value(train_data[example_no][0])
    elif data_set == "test":
        return _hypothesis_value(test_data[example_no][0])


def summation_of_cost_derivative(index, end=m):
    """
    Calculates the sum of cost function derivative
    :param index: index wrt derivative is being calculated
    :param end: value where summation ends, default is m, number of examples
    :return: Returns the summation of cost derivative
    Note: If index is -1, this means we are calculating summation wrt to biased parameter.
    """
    summation_value = 0
    for i in range(end):
        if index == -1:
            summation_value += _error(i)
        else:
            summation_value += _error(i) * train_data[i][0][index]
    return summation_value


def get_cost_derivative(index):
    """
    :param index: index of the parameter vector wrt to derivative is to be calculated
    :return: derivative wrt to that index
    Note: If index is -1, this means we are calculating summation wrt to biased parameter.
    """
    cost_derivative_value = summation_of_cost_derivative(index, m) / m
    return cost_derivative_value


def run_gradient_descent():
    global parameter_vector
    # Tune these values to set a tolerance value for predicted output
    absolute_error_limit = 0.000002
    relative_error_limit = 0
    j = 0
    while True:
        j += 1
        temp_parameter_vector = [0, 0, 0, 0]
        for i in range(0, len(parameter_vector)):
            cost_derivative = get_cost_derivative(i - 1)
            temp_parameter_vector[i] = (
                parameter_vector[i] - LEARNING_RATE * cost_derivative
            )
        if numpy.allclose(
            parameter_vector,
            temp_parameter_vector,
            atol=absolute_error_limit,
            rtol=relative_error_limit,
        ):
            break
        parameter_vector = temp_parameter_vector
    print(("Number of iterations:", j))


def test_gradient_descent():
    for i in range(len(test_data)):
        print(("Actual output value:", output(i, "test")))
        print(("Hypothesis output:", calculate_hypothesis_value(i, "test")))


if __name__ == "__main__":
    run_gradient_descent()
    print("\nTesting gradient descent for a linear hypothesis function.\n")
    test_gradient_descent()

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