Running this code we note an almost perfect agreement with the results from matrix inversion.
+# Using Autograd to calculate gradients using RMSprop and Stochastic Gradient descent
+# OLS example
+from random import random, seed
+import numpy as np
+import autograd.numpy as np
+import matplotlib.pyplot as plt
+from autograd import grad
+
+# Note change from previous example
+def CostOLS(y,X,theta):
+ return np.sum((y-X @ theta)**2)
+
+n = 10000
+x = np.random.rand(n,1)
+y = 2.0+3*x +4*x*x# +np.random.randn(n,1)
+
+X = np.c_[np.ones((n,1)), x, x*x]
+XT_X = X.T @ X
+theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y)
+print("Own inversion")
+print(theta_linreg)
+
+
+# Note that we request the derivative wrt third argument (theta, 2 here)
+training_gradient = grad(CostOLS,2)
+# Define parameters for Stochastic Gradient Descent
+n_epochs = 50
+M = 5 #size of each minibatch
+m = int(n/M) #number of minibatches
+# Guess for unknown parameters theta
+theta = np.random.randn(3,1)
+
+# Value for learning rate
+eta = 0.01
+# Value for parameter rho
+rho = 0.99
+# Including AdaGrad parameter to avoid possible division by zero
+delta = 1e-8
+for epoch in range(n_epochs):
+ Giter = np.zeros(shape=(3,3))
+ for i in range(m):
+ random_index = M*np.random.randint(m)
+ xi = X[random_index:random_index+M]
+ yi = y[random_index:random_index+M]
+ gradients = (1.0/M)*training_gradient(yi, xi, theta)
+ # Previous value for the outer product of gradients
+ Previous = Giter
+ # Accumulated gradient
+ Giter +=gradients @ gradients.T
+ # Scaling with rho the new and the previous results
+ Gnew = (rho*Previous+(1-rho)*Giter)
+ # Taking the diagonal only and inverting
+ Ginverse = np.c_[eta/(delta+np.sqrt(np.diagonal(Gnew)))]
+ # Hadamard product
+ update = np.multiply(Ginverse,gradients)
+ theta -= update
+print("theta from own RMSprop")
+print(theta)
+
+Running this code we note an almost perfect agreement with the results from matrix inversion.
+# Using Autograd to calculate gradients using RMSprop and Stochastic Gradient descent
+# OLS example
+from random import random, seed
+import numpy as np
+import autograd.numpy as np
+import matplotlib.pyplot as plt
+from autograd import grad
+
+# Note change from previous example
+def CostOLS(y,X,theta):
+ return np.sum((y-X @ theta)**2)
+
+n = 10000
+x = np.random.rand(n,1)
+y = 2.0+3*x +4*x*x# +np.random.randn(n,1)
+
+X = np.c_[np.ones((n,1)), x, x*x]
+XT_X = X.T @ X
+theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y)
+print("Own inversion")
+print(theta_linreg)
+
+
+# Note that we request the derivative wrt third argument (theta, 2 here)
+training_gradient = grad(CostOLS,2)
+# Define parameters for Stochastic Gradient Descent
+n_epochs = 50
+M = 5 #size of each minibatch
+m = int(n/M) #number of minibatches
+# Guess for unknown parameters theta
+theta = np.random.randn(3,1)
+
+# Value for learning rate
+eta = 0.01
+# Value for parameter rho
+rho = 0.99
+# Including AdaGrad parameter to avoid possible division by zero
+delta = 1e-8
+for epoch in range(n_epochs):
+ Giter = np.zeros(shape=(3,3))
+ for i in range(m):
+ random_index = M*np.random.randint(m)
+ xi = X[random_index:random_index+M]
+ yi = y[random_index:random_index+M]
+ gradients = (1.0/M)*training_gradient(yi, xi, theta)
+ # Previous value for the outer product of gradients
+ Previous = Giter
+ # Accumulated gradient
+ Giter +=gradients @ gradients.T
+ # Scaling with rho the new and the previous results
+ Gnew = (rho*Previous+(1-rho)*Giter)
+ # Taking the diagonal only and inverting
+ Ginverse = np.c_[eta/(delta+np.sqrt(np.diagonal(Gnew)))]
+ # Hadamard product
+ update = np.multiply(Ginverse,gradients)
+ theta -= update
+print("theta from own RMSprop")
+print(theta)
+
+# Using Autograd to calculate gradients using RMSprop and Stochastic Gradient descent
+# OLS example
+from random import random, seed
+import numpy as np
+import autograd.numpy as np
+import matplotlib.pyplot as plt
+from autograd import grad
+
+# Note change from previous example
+def CostOLS(y,X,theta):
+ return np.sum((y-X @ theta)**2)
+
+n = 10000
+x = np.random.rand(n,1)
+y = 2.0+3*x +4*x*x# +np.random.randn(n,1)
+
+X = np.c_[np.ones((n,1)), x, x*x]
+XT_X = X.T @ X
+theta_linreg = np.linalg.pinv(XT_X) @ (X.T @ y)
+print("Own inversion")
+print(theta_linreg)
+
+
+# Note that we request the derivative wrt third argument (theta, 2 here)
+training_gradient = grad(CostOLS,2)
+# Define parameters for Stochastic Gradient Descent
+n_epochs = 50
+M = 5 #size of each minibatch
+m = int(n/M) #number of minibatches
+# Guess for unknown parameters theta
+theta = np.random.randn(3,1)
+
+# Value for learning rate
+eta = 0.01
+# Value for parameter rho
+rho = 0.99
+# Including AdaGrad parameter to avoid possible division by zero
+delta = 1e-8
+for epoch in range(n_epochs):
+ Giter = np.zeros(shape=(3,3))
+ for i in range(m):
+ random_index = M*np.random.randint(m)
+ xi = X[random_index:random_index+M]
+ yi = y[random_index:random_index+M]
+ gradients = (1.0/M)*training_gradient(yi, xi, theta)
+ # Previous value for the outer product of gradients
+ Previous = Giter
+ # Accumulated gradient
+ Giter +=gradients @ gradients.T
+ # Scaling with rho the new and the previous results
+ Gnew = (rho*Previous+(1-rho)*Giter)
+ # Taking the diagonal only and inverting
+ Ginverse = np.c_[eta/(delta+np.sqrt(np.diagonal(Gnew)))]
+ # Hadamard product
+ update = np.multiply(Ginverse,gradients)
+ theta -= update
+print("theta from own RMSprop")
+print(theta)
+
+