3 Gradient Descent - Classification

4 Gradient descent with numpy



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%matplotlib inline
from fastai.learner import *

In this part of the lecture we explain Stochastic Gradient Descent (SGD) which is an optimization method commonly used in neural networks. We will illustrate the concepts with concrete examples.

Linear Regression problem

The goal of linear regression is to fit a line to a set of points.



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# Here we generate some fake data
def lin(a,b,x): return a*x+b

def gen_fake_data(n, a, b):
    x = s = np.random.uniform(0,1,n) 
    y = lin(a,b,x) + 0.1 * np.random.normal(0,3,n)
    return x, y

x, y = gen_fake_data(50, 3., 8.)



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plt.scatter(x,y, s=8); plt.xlabel("x"); plt.ylabel("y");

You want to find parameters (weights) $a$ and $b$ such that you minimize the error between the points and the line $a\cdot x + b$. Note that here $a$ and $b$ are unknown. For a regression problem the most common error function or loss function is the mean squared error.



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def mse(y_hat, y): return ((y_hat - y) ** 2).mean()

Suppose we believe $a = 10$ and $b = 5$ then we can compute y_hat which is our prediction and then compute our error.



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y_hat = lin(10,5,x)
mse(y_hat, y)



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def mse_loss(a, b, x, y): return mse(lin(a,b,x), y)



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mse_loss(10, 5, x, y)

So far we have specified the model (linear regression) and the evaluation criteria (or loss function). Now we need to handle optimization; that is, how do we find the best values for $a$ and $b$? How do we find the best fitting linear regression.

Gradient Descent

For a fixed dataset $x$ and $y$ mse_loss(a,b) is a function of $a$ and $b$. We would like to find the values of $a$ and $b$ that minimize that function.

Gradient descent is an algorithm that minimizes functions. Given a function defined by a set of parameters, gradient descent starts with an initial set of parameter values and iteratively moves toward a set of parameter values that minimize the function. This iterative minimization is achieved by taking steps in the negative direction of the function gradient.

Here is gradient descent implemented in PyTorch.



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# generate some more data
x, y = gen_fake_data(10000, 3., 8.)
x.shape, y.shape



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x,y = V(x),V(y)



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# Create random weights a and b, and wrap them in Variables.
a = V(np.random.randn(1), requires_grad=True)
b = V(np.random.randn(1), requires_grad=True)
a,b



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learning_rate = 1e-3
for t in range(10000):
    # Forward pass: compute predicted y using operations on Variables
    loss = mse_loss(a,b,x,y)
    if t % 1000 == 0: print(loss.data[0])
    
    # Computes the gradient of loss with respect to all Variables with requires_grad=True.
    # After this call a.grad and b.grad will be Variables holding the gradient
    # of the loss with respect to a and b respectively
    loss.backward()
    
    # Update a and b using gradient descent; a.data and b.data are Tensors,
    # a.grad and b.grad are Variables and a.grad.data and b.grad.data are Tensors
    a.data -= learning_rate * a.grad.data
    b.data -= learning_rate * b.grad.data
    
    # Zero the gradients
    a.grad.data.zero_()
    b.grad.data.zero_()

Nearly all of deep learning is powered by one very important algorithm: stochastic gradient descent (SGD). SGD can be seeing as an approximation of gradient descent (GD). In GD you have to run through all the samples in your training set to do a single itaration. In SGD you use only one or a subset of training samples to do the update for a parameter in a particular iteration. The subset use in every iteration is called a batch or minibatch.

Gradient Descent - Classification