In machine learning, gradient descent is an optimization technique used for computing the model parameters (coefficients and bias) for algorithms like linear regression, logistic regression, neural networks, etc. In this technique, we repeatedly iterate through the training set and update the model parameters in accordance with the gradient of the error with respect to the training set. Depending on the number of training examples considered in updating the model parameters, we have 3-types of gradient descents:
- Batch Gradient Descent: Parameters are updated after computing the gradient of the error with respect to the entire training set
- Stochastic Gradient Descent: Parameters are updated after computing the gradient of the error with respect to a single training example
- Mini-Batch Gradient Descent: Parameters are updated after computing the gradient of the error with respect to a subset of the training set
| Batch Gradient Descent | Stochastic Gradient Descent | Mini-Batch Gradient Descent |
|---|---|---|
| Since the entire training data is considered before taking a step in the direction of gradient, therefore it takes a lot of time for making a single update. | Since only a single training example is considered before taking a step in the direction of gradient, we are forced to loop over the training set and thus cannot exploit the speed associated with vectorizing the code. | Since a subset of training examples is considered, it can make quick updates in the model parameters and can also exploit the speed associated with vectorizing the code. |
| It makes smooth updates in the model parameters | It makes very noisy updates in the parameters | Depending upon the batch size, the updates can be made less noisy – greater the batch size less noisy is the update |
Thus, mini-batch gradient descent makes a compromise between the speedy convergence and the noise associated with gradient update which makes it a more flexible and robust algorithm.
Convergence in BGD, SGD & MBGD
Mini-Batch Gradient Descent: Algorithm-
Let theta = model parameters and max_iters = number of epochs. for itr = 1, 2, 3, …, max_iters: for mini_batch (X_mini, y_mini):
- Forward Pass on the batch X_mini:
- Make predictions on the mini-batch
- Compute error in predictions (J(theta)) with the current values of the parameters
- Backward Pass:
- Compute gradient(theta) = partial derivative of J(theta) w.r.t. theta
- Update parameters:
- theta = theta – learning_rate*gradient(theta)
Below is the Python Implementation:
Step #1: First step is to import dependencies, generate data for linear regression, and visualize the generated data. We have generated 8000 data examples, each having 2 attributes/features. These data examples are further divided into training sets (X_train, y_train) and testing set (X_test, y_test) having 7200 and 800 examples respectively.
Python3
# importing dependenciesimport numpy as npimport matplotlib.pyplot as plt# creating datamean = np.array([5.0, 6.0])cov = np.array([[1.0, 0.95], [0.95, 1.2]])data = np.random.multivariate_normal(mean, cov, 8000)# visualising dataplt.scatter(data[:500, 0], data[:500, 1], marker='.')plt.show()# train-test-splitdata = np.hstack((np.ones((data.shape[0], 1)), data))split_factor = 0.90split = int(split_factor * data.shape[0])X_train = data[:split, :-1]y_train = data[:split, -1].reshape((-1, 1))X_test = data[split:, :-1]y_test = data[split:, -1].reshape((-1, 1))print(& quot Number of examples in training set= % d & quot % (X_train.shape[0]))print(& quot Number of examples in testing set= % d & quot % (X_test.shape[0])) |
Output:
Number of examples in training set = 7200 Number of examples in testing set = 800
Step #2: Next, we write the code for implementing linear regression using mini-batch gradient descent. gradientDescent() is the main driver function and other functions are helper functions used for making predictions – hypothesis(), computing gradients – gradient(), computing error – cost() and creating mini-batches – create_mini_batches(). The driver function initializes the parameters, computes the best set of parameters for the model, and returns these parameters along with a list containing a history of errors as the parameters get updated.
Example
Python3
# linear regression using "mini-batch" gradient descent# function to compute hypothesis / predictionsdef hypothesis(X, theta): return np.dot(X, theta)# function to compute gradient of error function w.r.t. thetadef gradient(X, y, theta): h = hypothesis(X, theta) grad = np.dot(X.transpose(), (h - y)) return grad# function to compute the error for current values of thetadef cost(X, y, theta): h = hypothesis(X, theta) J = np.dot((h - y).transpose(), (h - y)) J /= 2 return J[0]# function to create a list containing mini-batchesdef create_mini_batches(X, y, batch_size): mini_batches = [] data = np.hstack((X, y)) np.random.shuffle(data) n_minibatches = data.shape[0] // batch_size i = 0 for i in range(n_minibatches + 1): mini_batch = data[i * batch_size:(i + 1)*batch_size, :] X_mini = mini_batch[:, :-1] Y_mini = mini_batch[:, -1].reshape((-1, 1)) mini_batches.append((X_mini, Y_mini)) if data.shape[0] % batch_size != 0: mini_batch = data[i * batch_size:data.shape[0]] X_mini = mini_batch[:, :-1] Y_mini = mini_batch[:, -1].reshape((-1, 1)) mini_batches.append((X_mini, Y_mini)) return mini_batches# function to perform mini-batch gradient descentdef gradientDescent(X, y, learning_rate=0.001, batch_size=32): theta = np.zeros((X.shape[1], 1)) error_list = [] max_iters = 3 for itr in range(max_iters): mini_batches = create_mini_batches(X, y, batch_size) for mini_batch in mini_batches: X_mini, y_mini = mini_batch theta = theta - learning_rate * gradient(X_mini, y_mini, theta) error_list.append(cost(X_mini, y_mini, theta)) return theta, error_list |
Calling the gradientDescent() function to compute the model parameters (theta) and visualize the change in the error function.
Python3
theta, error_list = gradientDescent(X_train, y_train)print("Bias = ", theta[0])print("Coefficients = ", theta[1:])# visualising gradient descentplt.plot(error_list)plt.xlabel("Number of iterations")plt.ylabel("Cost")plt.show() |
Output: Bias = [0.81830471] Coefficients = [[1.04586595]]
Step #3: Finally, we make predictions on the testing set and compute the mean absolute error in predictions.
Python3
# predicting output for X_testy_pred = hypothesis(X_test, theta)plt.scatter(X_test[:, 1], y_test[:, ], marker='.')plt.plot(X_test[:, 1], y_pred, color='orange')plt.show()# calculating error in predictionserror = np.sum(np.abs(y_test - y_pred) / y_test.shape[0])print(& quot Mean absolute error = & quot , error) |
Output:
Mean absolute error = 0.4366644295854125
The orange line represents the final hypothesis function: theta[0] + theta[1]*X_test[:, 1] + theta[2]*X_test[:, 2] = 0
