Parameter Updation
Updating the weight and bias by subtracting the multiplication of learning rates and their respective gradients.
w = w - (learning_rate * (dJ/dw)) b = b - (learning_rate * (dJ/db))
Python Implementation for Gradient Descent
In the implementation part, we will be writing two functions, one will be the cost functions that take the actual output and the predicted output as input and returns the loss, the second will be the actual gradient descent function which takes the independent variable, target variable as input and finds the best fit line using gradient descent algorithm. The iterations, learning_rate, and stopping threshold are the tuning parameters for the gradient descent algorithm and can be tuned by the user. In the main function, we will be initializing linearly related random data and applying the gradient descent algorithm on the data to find the best fit line. The optimal weight and bias found by using the gradient descent algorithm are later used to plot the best fit line in the main function. The iterations specify the number of times the update of parameters must be done, the stopping threshold is the minimum change of loss between two successive iterations to stop the gradient descent algorithm.
Python3
# Importing Librariesimport numpy as npimport matplotlib.pyplot as pltdef mean_squared_error(y_true, y_predicted): # Calculating the loss or cost cost = np.sum((y_true-y_predicted)**2) / len(y_true) return cost# Gradient Descent Function# Here iterations, learning_rate, stopping_threshold# are hyperparameters that can be tuneddef gradient_descent(x, y, iterations = 1000, learning_rate = 0.0001, stopping_threshold = 1e-6): # Initializing weight, bias, learning rate and iterations current_weight = 0.1 current_bias = 0.01 iterations = iterations learning_rate = learning_rate n = float(len(x)) costs = [] weights = [] previous_cost = None # Estimation of optimal parameters for i in range(iterations): # Making predictions y_predicted = (current_weight * x) + current_bias # Calculating the current cost current_cost = mean_squared_error(y, y_predicted) # If the change in cost is less than or equal to # stopping_threshold we stop the gradient descent if previous_cost and abs(previous_cost-current_cost)<=stopping_threshold: break previous_cost = current_cost costs.append(current_cost) weights.append(current_weight) # Calculating the gradients weight_derivative = -(2/n) * sum(x * (y-y_predicted)) bias_derivative = -(2/n) * sum(y-y_predicted) # Updating weights and bias current_weight = current_weight - (learning_rate * weight_derivative) current_bias = current_bias - (learning_rate * bias_derivative) # Printing the parameters for each 1000th iteration print(f"Iteration {i+1}: Cost {current_cost}, Weight \ {current_weight}, Bias {current_bias}") # Visualizing the weights and cost at for all iterations plt.figure(figsize = (8,6)) plt.plot(weights, costs) plt.scatter(weights, costs, marker='o', color='red') plt.title("Cost vs Weights") plt.ylabel("Cost") plt.xlabel("Weight") plt.show() return current_weight, current_biasdef main(): # Data X = np.array([32.50234527, 53.42680403, 61.53035803, 47.47563963, 59.81320787, 55.14218841, 52.21179669, 39.29956669, 48.10504169, 52.55001444, 45.41973014, 54.35163488, 44.1640495 , 58.16847072, 56.72720806, 48.95588857, 44.68719623, 60.29732685, 45.61864377, 38.81681754]) Y = np.array([31.70700585, 68.77759598, 62.5623823 , 71.54663223, 87.23092513, 78.21151827, 79.64197305, 59.17148932, 75.3312423 , 71.30087989, 55.16567715, 82.47884676, 62.00892325, 75.39287043, 81.43619216, 60.72360244, 82.89250373, 97.37989686, 48.84715332, 56.87721319]) # Estimating weight and bias using gradient descent estimated_weight, estimated_bias = gradient_descent(X, Y, iterations=2000) print(f"Estimated Weight: {estimated_weight}\nEstimated Bias: {estimated_bias}") # Making predictions using estimated parameters Y_pred = estimated_weight*X + estimated_bias # Plotting the regression line plt.figure(figsize = (8,6)) plt.scatter(X, Y, marker='o', color='red') plt.plot([min(X), max(X)], [min(Y_pred), max(Y_pred)], color='blue',markerfacecolor='red', markersize=10,linestyle='dashed') plt.xlabel("X") plt.ylabel("Y") plt.show() if __name__=="__main__": main() |
Output:
Iteration 1: Cost 4352.088931274409, Weight 0.7593291142562117, Bias 0.02288558130709
Iteration 2: Cost 1114.8561474350017, Weight 1.081602958862324, Bias 0.02918014748569513
Iteration 3: Cost 341.42912086804455, Weight 1.2391274084945083, Bias 0.03225308846928192
Iteration 4: Cost 156.64495290904443, Weight 1.3161239281746984, Bias 0.03375132986012604
Iteration 5: Cost 112.49704004742098, Weight 1.3537591652024805, Bias 0.034479873154934775
Iteration 6: Cost 101.9493925395456, Weight 1.3721549833978113, Bias 0.034832195392868505
Iteration 7: Cost 99.4293893333546, Weight 1.3811467575154601, Bias 0.03500062439068245
Iteration 8: Cost 98.82731958262897, Weight 1.3855419247507244, Bias 0.03507916814736111
Iteration 9: Cost 98.68347500997261, Weight 1.3876903144657764, Bias 0.035113776874486774
Iteration 10: Cost 98.64910780902792, Weight 1.3887405007983562, Bias 0.035126910596389935
Iteration 11: Cost 98.64089651459352, Weight 1.389253895811451, Bias 0.03512954755833985
Iteration 12: Cost 98.63893428729509, Weight 1.38950491235671, Bias 0.035127053821718185
Iteration 13: Cost 98.63846506273883, Weight 1.3896276808137857, Bias 0.035122052266051224
Iteration 14: Cost 98.63835254057648, Weight 1.38968776283053, Bias 0.03511582492978764
Iteration 15: Cost 98.63832524036214, Weight 1.3897172043139192, Bias 0.03510899846107016
Iteration 16: Cost 98.63831830104695, Weight 1.389731668997059, Bias 0.035101879159522745
Iteration 17: Cost 98.63831622628217, Weight 1.389738813163012, Bias 0.03509461674147458
Estimated Weight: 1.389738813163012
Estimated Bias: 0.03509461674147458
Cost function approaching local minima
The best fit line obtained using gradient descent
How to implement a gradient descent in Python to find a local minimum ?
Gradient Descent is an iterative algorithm that is used to minimize a function by finding the optimal parameters. Gradient Descent can be applied to any dimension function i.e. 1-D, 2-D, 3-D. In this article, we will be working on finding global minima for parabolic function (2-D) and will be implementing gradient descent in python to find the optimal parameters for the linear regression equation (1-D). Before diving into the implementation part, let us make sure the set of parameters required to implement the gradient descent algorithm. To implement a gradient descent algorithm, we require a cost function that needs to be minimized, the number of iterations, a learning rate to determine the step size at each iteration while moving towards the minimum, partial derivatives for weight & bias to update the parameters at each iteration, and a prediction function.
Till now we have seen the parameters required for gradient descent. Now let us map the parameters with the gradient descent algorithm and work on an example to better understand gradient descent. Let us consider a parabolic equation y=4x2. By looking at the equation we can identify that the parabolic function is minimum at x = 0 i.e. at x=0, y=0. Therefore x=0 is the local minima of the parabolic function y=4x2. Now let us see the algorithm for gradient descent and how we can obtain the local minima by applying gradient descent:
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